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Figures 57.1 and 57.2 originally appeared on pages 79 and 80 in R. Piessens, E. de 
Doncker-Kapenga, C.W. Uherhuber, and D.K. Kahaner, Quadpack, Springer-Verlag, 1983. 
Reprinted (.'ourtesy of Springer-Verlag. 
Library of Congress Cataloging-in-Publication Data 
Zwillinger, Daniel, 1957-
Handbook of integration I Daniel Zwillinger. 
p. em. 
Includes bihiliographical references and index. 
ISBN 0-86720-293-9 
1. Numerical integnltion. I. Title. 
QA299.3.Z85 1992 
515'.43-dc20 
Printed in the United States of America 
92-14050 
CIP 
96 95 94 93 92 10987654321 
Table of Contents 
Preface 
Introduction . . . . 
How to Use This Book 
I Applications of Integration 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
Differential Equations: Integral Representations 
Differential Equations: Integral Transforms 
Extremal Problems 
Function Representation 
Geometric Applications 
MIT Integration Bee 
Probability ..... 
Summations: Combinatorial 
Summations: Other . . . 
Zeros of Functions 
Miscellaneous Applications 
II Concepts and Definitions 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
Definitions . . . . 
Integral Definitions 
Caveats ..... 
Changing Order of Integration 
Convergence of Integrals 
Exterior Calculus . . . 
Feynman Diagrams .. 
Finite Part 0f Integrals 
Fractional Integration 
Liouville Theory 
Mean Value Theorems 
Path Integrals 
Principal Value Integrals 
Transforms: To a Finite Interval 
Transforms: Multidimensional Integrals 
Transforms: Miscellaneous . . . . . . 
v 
IX 
xi 
.Xlll 
1 
6 
14 
20 
24 
28 
30 
31 
34 
40 
45 
47 
51 
58 
61 
64 
67 
70 
73 
75 
79 
83 
86 
92 
95 
97 
103 
vi 
III Exact Analytical Methods 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
Change of Variable 
Computer Aided Solution 
Contour Integration . . . 
Convolution Techniques . 
Differentiation and Integration 
Dilogarithms . . . 
Elliptic Integrals 
Frullanian Integrals . 
Functional Equations 
Integration by Parts 
Line and Surface Integrals 
Look Up Technique . . . 
Special Integration Techniques 
Stochastic Integration 
Tables of Integrals ..... 
IV Approximate Analytical Methods 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
Asymptotic Expansions . . . . . . . . 
Asymptotic Expansions: Multiple Integrals 
Continued Fractions 
Integral Inequalities 
Integration by Parts 
Interval Analysis 
Laplace's Method 
Stationary Phase 
Steepest Descent 
Approximations: Miscellaneous 
V Numerical Methods: Concepts 
53 Introduction to Numerical Methods 
54 Numerical Definitions 
55 Error Analysis ., 
56 Romberg Integration / Richardson Extrapolation 
57 Software Libraries: Introduction 
58 Software Libraries: Taxonomy 
59 Software Libraries: Excerpts from G AMS 
60 Testing Quadrature Rules 
61 Truncating an Infinite Interval 
109 
117 
129 
140 
142 
145 
148 
157 
160 
162 
164 
170 
181 
186 
190 
195 
199 
203 
205 
215 
218 
221 
226 
230 
240 
243 
244 
246 
250 
254 
258 
260 
272 
275 
Table of Contents vii 
VI Numerical Methods: Techniques 
62 Adaptive Quadrature 277 
63 Clenshaw-Curtis Rules 281 
64 Compound Rules 283 
65 Cubic Splines 285 
66 Using Derivative Information 287 
67 Gaussian Quadrature 289 
68 Gaussian Quadrature: Generalized 292 
69 Gaussian Quadrature: Kronrod's Extension 298 
70 Lattice Rules 300 
71 Monte Carlo Method 304 
72 N umber Theoretic Methods 312 
73 Parallel Computer Methods 315 
74 Polyhedral Symmetry Rules 316 
75 Polynomial Interpolation 319 
76 Product Rules 323 
77 Recurrence Relations 325 
78 Symbolic Methods 329 
79 Tschebyscheff Rules 332 
80 Wozniakowski's Method 333 
81 Tables: Numerical Methods 337 
82 Tables: Formulas for Integrals 340 
83 Tables: Numerically Evaluated Integrals 348 
Mathematical Nomenclature 351 
Index 353 
I Applications of Integration 6 
2. Differential Equations: 
Integral Transforms 
Applicable to Linear differential equations. 
Idea 
In order to solve a linear differential equation, it is sometimes easier to 
transform the equation to some “space,” solve the equation in that “space,” 
and then transform the solution back. 
Procedure 
Given a linear differential equation, multiply the equation by a kernel 
and integrate over a specified region (see Table 2.1 and Table 2.2 for a 
listing of common kernels and limits of integration). Use integration by 
parts to obtain an equation for the transform of the dependent variable. 
You will have used the “correct” transform (i.e., you have chosen the 
correct kernel and limits) if the boundary conditions given with the original 
equation have been utilized. Now solve the equation for the transform of 
the dependent variable. From this, obtain the solution by multiplying by 
the inverse kernel and performing another integration. Table 2.1 and Table 
2.2 also list the inverse kernel. 
Example 
Suppose we have the boundary value problem for y = y(x) 
yxx + y = 1, 
y(0) = 0, y(1) = 0. 
(2.1 .a-c) 
Since the solution vanishes at both of the endpoints, we suspect that a 
finite sine transform might be a useful transform to try. Define the finite 
sine transform of y(x) to be x(t), so that 
(See “finite sine transform-2” in Table 2.1). Now multiply equation (2.1.a) 
by sintx and integrate with respect to x from 0 to 1. This results in 
Jdlyxx(x)sintxdx+ y(x)sin<xdx = sintxdx. (2.3) 
Jdl Jdl 
If we integrate the first term in (2.3) by parts, twice, we obtain 
x=l x = l 
yzx (x) sin tx dx = yx (2) sin txl - ty(x) cos txl 
- t 2 Jdl y(x) sin tx dx. (2.4) I’ x = o x=o 
8 I Applications of Integration 
Table 2.1 Different transform pairs of the form 
Finite cosine transform - 1, (see Miles [17], page 86) here 1 and h are arbitrary, 
and the {&} satisfy & tanckl = h. 
Finite cosine transform - 2, (see Butkov [3], page 161) this is the last 
transform with h = 0, Z = 1, so that <k = 0, 7r, 27r,. . . . 
Finite sine transform - 1, (see Miles [17], page 86) here 1 and h are arbitrary, 
and the {&} satisfy <k cot(&Z) = -h. 
Finite sine transform - 2, (see Butkov [3], page 161) this is the last transform 
with h = 0, I = 1, so that & = 0,7rr,2n,. . . . 
Finite Hankel transform - 1, (see Tranter [24], page 88) here n is arbitrary 
and the {&} are positive and satisfy Jn(<k) = 0. 
Finite Hankel transform - 2, (see Miles [17], page 86) here n and h are 
arbitrary and the {Jk} are positive and satisfy <k JA(u&) + hJn(u<k) = 0. 
2. Differential Equations: Integral Transforms 11 
Weber formula, (See Titchmarsh [23], page 75) 
Weierstrass transform, (See Hirschman and Widder [ll], Chapter 8) 
Notes 
There are many tables of transforms available (See Bateman [3] or Magnus, 
Oberhettinger, and Soni [16]). It is generally easier to look up a transform 
than to compute it. 
Transform techniques may also be used with Systems of linear equations. 
Transforms may also be evaluated numerically. There are many results on 
how to compute the more popular transforms numerically, like the Laplace 
transform. See, for example, Strain [22]. 
The finite Hankel transforms are useful for differential equations that contain 
the Operator LH[u] and the Legendre transform is useful for differential 
equations that contain the Operator LL[u], where 
ur n2 a 2 du LH[u] = U r i + - - Fu and LL[u] = - ((1 - r )&) . ar 
For example, the Legendre transform of LL[u] is simply -&(& + l)w(<k). 
Integral transforms are generally createdfor solving a specific differential 
equation with a specific class of boundary conditions. The Mathieu inte- 
gral transform (See Inayat-Hussain [12]) has been constructed for the two- 
dimensional Helmholtz equation in elliptic-cylinder Coordinates. 
Integral transforms can also be constructed by integrating the Green’s func- 
tion for a Sturm-Liouville eigenvalue Problem. See Zwillinger [25] for details. 
Note that many of the transforms in Table 2.1 and Table 2.2 do not have a 
Standard form. In the Fourier transform, for example, the two 6 terms 
might not be symmetrically placed as we have shown them. Also, a small 
Variation of the K-transform is known as the Meijer transform (See Ditkin 
and Prudnikov [8], page 75). 
If a function f (z ,y) has radial symmetry, then a Fourier transform in both 
z and y is equivalent to a Hankel transform of f ( r ) = f ( z , y ) , where r2 = 
z2 + y2. See Sneddon [20], pages 79-83. 
12 I Applications of Integration 
[9] Two transform pairs that are continuous in one variable and discrete in the 
other variable, on an infinite interval, are the Hermite transform 
where H n ( x ) is the n-th Hermite polynomial, and the Laguerre transform 
where L g ( x ) is the Laguerre polynomial of degree n, and Q 2 0. See 
Haimo [ l O ] for details. 
[ l O ] Classically, the Fourier transform of a function only exists if the function 
being transformed decays quickly enough at f00. The Fourier transform 
can be extended, though, to handle generalized functions. For example, the 
Fourier transform of the n-th derivative of the delta function is given by 
Another way to approach the Fourier transform of functions that do 
not decay quickly enough at either 00 or -00 is to use the one-sided Fourier 
trunsforms. See Chester [6] for details. 
[ll] Many of the transforms listed generalize naturally to n dimensions. For 
examde. in n dimensions we have: 
F [ S ( " ) ( t ) ] = (Zu)". 
I , 
U ( ( ) = (27r)-n/2 Ln e g ' x u ( x ) d x , 
u(x) = ( 2 ~ ) - ~ / ~ ln e - g ' x v ( t ) d t . (A) Fourier transform: 
(B) Hilbert transform (See Bitsadze [4]): 
[12] Apelblat [2] has found that repeated use of integral transforms can lead to 
the simplification of some infinite integrals. For example, let F,(y) denote 
the Fourier sine transform of the function f ( x ) , F,(y) = Jom f ( x ) s i n y x d x . 
Taking the Laplace transform of this results in 
G(s) = .bm e-51/ { JI" f ( x ) sin yx d x } dy 
00 
where the Order of integration has been changed and then the inner integral 
evaluated. For some functions f ( x ) it may be possible to find the corre- 
sponding F, (Y) and G(s) using comprehensive tables of integral transforms. 
2. Differential Equat ions: Integral Transforms 13 
Equating this to the expression in (2.6) may result in a definite integral hard 
to evaluate in other ways. 
1 As a simple example of the technique, consider using f(z) = 
x(u2 + x”>’ 
With this we can find F,(y) = 
of this, and equating the result to (2.6), we have found the integral 
(1 - e-ag). Taking the Laplace transform 
[13] Carson’s integral is the integral transformation Q ( p ) = p ~ o O O e-”‘.f(t) dt. See 
[14] A transform pair that is continuous in each variable, on a finite interval, is 
Iyanaga and Kawada [8]. 
the finite Hilbert transform 
where C is an arbitrary constant, and the integrals are principal value 
integrals. See Sneddon [20], page 467, for details. 
[15] Note that, for the Hilbert transform, the integrals in Table 2.2 are principal 
value integrals. 
R e ferences 
[l] 
[2] 
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 
National Bureau of Standards, Washington, DC, 1964, pages 1019-1030. 
A. Apelblat, “Repeating Use of Integral Transforms-A New Method for 
Evaluation of Some Infinite Integrals,” IMA J. Appl. Mathematics, 27, 1981, 
pages 481-496. 
Staff of the Bateman Manuscript Project, A. Erdklyi (ed.), Tables of Integral 
Transforms, in 3 volumes, McGraw-Hill Book Company, New York, 1954. 
A. V. Bitsadze, “The Multidimensional Hilbert Transform,” Soviet Math. 
Dokl., 35, No. 2, 1987, pages 390-392. 
R. N. Bracewell, The Hartley Zhnsform, Oxford University Press, New 
York, 1986. 
C. R. Chester, Techniques in Partial Difrerential Equations, McGraw-Hill 
Book Company, New York, 1970. 
B. Davies, Integral Transforms und Their Applications - Second Edition, 
Springer-Verlag, New York, 1985. 
V. A. Ditkin and A. P. Prudnikov, Integral Transforms und Operational 
Calculus, translated by D. E. Brown, English translation edited by I. N. 
Sneddon, Pergamon Press, New York, 1965. 
[9] H.-J. Glaeske, “Operational Properties of a Generalized Hermite Transfor- 
mation,” Aequationes Mathematicae, 32, 1987, pages 155-170. 
[ lO] D. T. Haimo, “The Dual Weierstrass-Laguerre Transform,” Trans. AMS, 
290, No. 2, August 1985, pages 597-613. 
[ll] I. I. Hirschman and D. V. Widder, The Convolution Transform, Princeton 
University Press, Princeton, NJ , 1955. 
[3] 
[4] 
[5] 
[6] 
[7] 
[8] 
14 I Applications of Integration 
[12] A. A. Inayat-Hussain, “Mathieu Integral Transforms,” J. Math. Physics, 32, 
No. 3, March 1991, pages 669-675. 
[13] S. Iyanaga and Y. Kawada, Encyclopedic Dictionary of Mathematics, MIT 
Press, Cambridge, MA, 1980. 
[14] D. S. Jones, “The Kontorovich-Lebedev Transform,” J. Inst. Maths. Ap- 
plics, 26, 1980, pages 133-141. 
[15] 0. I. Marichev, Handbook of Integral l’ransforms of Higher Transcendental 
Functions: Theory und Algorithmic Tables, translated by L. W. Longdon, 
Halstead Press, John Wiley & Sons, New York, 1983. 
[16] W. Magnus, F. Oberhettinger, and R. P. Soni, Fomnulus und Theorems for 
the Special Functions of Mathematical Physics, Springer-Verlag, New York, 
1966. 
[17] J. W. Miles, Integral Transforms in Applied Mathematics, Cambridge Uni- 
versity Press, 1971. 
[18] C. Nasim, “The Mehler-Fock Transform of General Order and Arbitrary 
Index and Its Inversion,’’ Int. J. Math. & Math. Sci., 7, No. 1, 1984, pages 
[19] F. Oberhettinger and T. P. Higgins, Tables of Lebedev, Mehler, und Gen- 
eralzzed Mehler Transforms, Mathematical Note No. 246, Boeing Scientific 
Research Laboratories, October 1961. 
[20] I. N. Sneddon, The Use of Integral Z’runsforms, McGraw-Hill Book Com- 
pany, New York, 1972. 
[21] I. Stakgold, Green’s Functions und Boundary Vulue Problems, John Wiley 
& Sons, New York, 1979. 
[22] J. Strain, “A Fast Laplace Transform Based on Laguerre Functions,” hlath. 
of Comp., 58, No. 197, January 1992, pages 275-283. 
[23] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order 
Diflerential Equations, Clarendon Press, Oxford, 1946. 
[24] C. J. Tranter, Integral l’ransforms in Muthematical Physics, Methuen & Co. 
Ltd., London, 1966. 
[25] D. Zwillinger, Handbook of Diferential Equations, Academic Press, New 
York, Second Edition, 1992. 
171-180. 
3. Extrernal Problems 
Applicable to 
integral. 
Finding a function that maximizes (or minimizes) an 
Yields 
A differential equation for the critical function. 
3. Extremal Problems 15 
Procedure 
Given the functional 
R 
where the Operator L ( ) is a linear or nonlinear function of its arguments, 
how can u(x) be determined so that J[u] is critical (i.e., either a maximum 
or a minimum)? 
The variational principle that is most often used is SJ = 0, which 
states that the integral J[u] should be stationary with respect to small 
changes in u(x). If we let h(x) be a “small,” continuously differentiable 
function, then we can form 
J[u+h]-J[u] = // [L(x,azj)(u(x) + h(x)) - L(X,~~,)U(X) dx. (3.2) 1 
R 
By integration by Parts, (3.2) can often be written as 
plus some 
J [ u + h] - 
J [ u + h] - J[u] = N ( x , azj)u(x) dx + JJ R 
boundary terms. The variational principle 
J[u] vanishes to leading Order, or that 
N(x,azj)u(x) = 0. 
Equation (3.3) is called the first Variation of (3.1), or 
requires that 6J := 
(3.3) 
t he Euler-Lagrange 
equation corresponding to (3.1).(This is also called the Euler equation.) 
A functional in the form of (3.1) determines an Euler-Lagrange equation. 
Conversely, given an Euler-Lagrange equat ion, a corresponding funct ional 
can sometimes be obtained. 
Many approximate and numerical techniques for differential equations 
utilize the functional associated with a given System of Euler-Lagrange 
equations. For example, both the Rayleigh-Ritz method and the finite 
element method create (in principle) integrals that are then analyzed (See 
Zwillinger [ 5 ] ) . 
The following collection of examples assume that the dependent vari- 
able in the given differential equation has natural boundary conditions. If 
the dependent variable did not have these specific boundary conditions, 
then the boundary terms that were discarded in going from (3.2) to (3.3) 
would have to be satisfied in addition to the Euler-Lagrange equation. 
3. Extremal Problems 17 
Example 2 
The Euler-Lagrange equation for the functional 
where y = y(x) is 
---(-)+--&($)-...+(- dF d dF d2 
dy dx dy’ 
For this equation the natural boundary conditions are given by 
Example 3 
The Euler-Lagrange equation for the functional 
where U = u(x, Y) is 
d2 d2 +- (-) + ay2 (E) = 0. 
dxdy duxy dUYY 
Example 4 
The Euler-Lagrange equation for t he functional 
J[u]=// [ a ( ~ ) 2 + b ( ~ ) 2 + c u 2 + 2 f u 1 dxdy, 
R 
which is a Special case of (3.6), is: - ;x (.E) + -& ( b $ ) -CU = f. 
18 
J[u] = F ( z , U , U’) dz - 91 (z, U ) + 92(z, U ) 
I Applications of Integration 
, 
Example 5 
adjoint form) 
For the 2m-th Order ordinary differential equation (in formally self- 
U‘ -k au = 0, u‘+ßu 
a corresponding functional is 
= 0, 
J [ ~ ] = Ix2 [f ( u ‘ ) ~ - ~ ( z ) u ] dz + - 7 aU2 
2 1 x=x2 x=x1 
3. Extremal Problems 19 
[3] This technique can be used in higher dimensions. For example, consider the 
functional 
R 
where d / d a and d / d n are partial differential Operators in the directions of 
the tangent and normal to the curve dR. Necessary conditions for J [ u ] to 
have a minimum are the Euler-Lagrange equations (given in (3.6)) together 
with the boundary conditions: 
d dG d2 dG +G, - -- +--- - 0, aa du, da2 du,, 
xoyo = 0, x;+- - +--Y:+- ou, duxx du„ auxy 
dF dG dF d F 
where x, = d x / d a and Y, = dy/da. See Mitchell and Wait [3] for details. 
Re ferences 
[ 11 
[2] 
[3] 
[4] 
[5] 
E. Butkov, Mathematical Physics, Addison-Wesley Publishing Co., Reading, 
MA, 1968, pages 573-588. 
L. V. Kantorovich and V. I. Krylov, Approximate Methods of Higher Anal- 
ysis, Interscience Publishers, New York, 1958, Chapter 4, pages 241-357. 
A. R. Mitchell and R. Wait, The Finite Element Method in Diferential 
Equations, Wiley, New York, 1977, pages 27-31. 
H. Rund, The Hamilton-Jacobi Theory in the Calculus of Variations, D. Van 
Nostrand Company, Inc., New York, 1966. 
D. Zwillinger, Handbook of Diflerential Equations, Academic Press, New 
York, Second Edition, 1992. 
20 I Applicat ions of Integration 
4. Function Representation 
Idea 
Certain integrals can be used to represent functions. 
This section contains several different representational theorems. Each 
Procedure 
has found many important applications in the literature. 
Bochner-Martinelli Representation 
Let f be a holomorphic function in a domain D c C", with 
piecewise smooth boundary dD, and let f be continuous in 
its closure D. Then we have the representation 
h 
where dCi = dCl A dC1 A - - - A [dTi] A - - - A dTn A dCn, and [dci] 
means that the term dTi is to be omitted. 
For n = 1, this is identical to the Cauchy representation. Another way 
to write this result is as follows: 
Let H ( G ) be the ring of holomorphic functions in G. Let Gj 
be a domain in the zj-plane with piecewise smooth boundary 
Cj. If f E H ( G ) (where G := nj"=, Gj) is continuous on G, 
then 
For details see Krantz [5] or Iyanaga and Kawada [6] (page 101). 
Cauchy Representation 
Cauchy's integral formula states that if a domain D is bounded by 
a finite Union of simple closed curves I?, and f is analytic within D and 
across r, then 
for < E D. (See page 129 for several applications of this formula.) 
integral formula 
If D is the disk 1.1 < R, then Cauchy's theorem becomes Poisson's 
d4. 
R2 - r2 
R2 + r2 - 2Rr cos(8 - 4) 
4. E'unct ion Represent at ion 21 
There is an analogous formula, called Villat's integral formula, when D is 
an annulus. See Iyanaga and Kawada [6], page 636. 
There are also extensions of this formula when f (z) is not analytic. 
In terms of the differential Operator & = $ (dz + dg), the Cauchy-Green 
formula is 
for < E D. This formula is valid whenever f is smooth enough for the 
derivative & to make sense. If f is analytic, then the Cauchy-Riemann 
equations hold; these equations are equivalent to & = 0. See Khavinson [7] 
for details. 
Green's Representation Theorems 
0 
0 
See 
Three dimensions: If 4 and V24 are defined within a volume V bounded 
by a simple closed surface S , p is an interior point of V , and n 
represents the outward unit normal, then 
(4.1) 
Note that if q5 is harmonic (i.e., V24 = 0), then the right-hand side of 
(4.1) simplifies. 
Two dimensions: If 4 and V2+ are defined within a planar region S 
bounded by a simple closed curve C, p is an interior point of S , and 
nq represents the outward unit normal at the point q, then 
n dimensions: If 4 and its second derivatives are defined within a 
region R in Rn bounded by the surface E, and nq represents the 
outward unit normal at the point q, then for Points p not on the 
surface C we have (if n > 3) 
where an = 2~"/~//r(n/2> is the area of a unit sphere in Rn. 
Gradshteyn and Rvzhik 121, 10.717, pages 1089-1090 for details. 
22 I Applications of Integration 
Herglotz's Integral Representation 
representation. It states: 
The Herglotz integral representation is based on Poisson's integral 
Let f(z) be holomorphic in IzI < R with positive real part. 
for IzI < R, where p ( 4 ) is a monotonic increasing real-valued 
function with total Variation unity. This function is deter- 
mined uniquely, up to an additive constant, by f(z). 
See Iyanaga and Kawada [6], page 161. Another Statement of this 
integral representation is (see Hazewinkel [3], page 124): 
Let f (z) be regular in the unit disk D = { z I IzI < 1)) and 
assume that it has a positive real part (i.e., Ref(z) < 0 ) , 
then f(z) can be represented as 
where the imaginary part of c is zero. Here p is a positive 
measure concentrated on the circle {t I = 1). 
Parametric Representation of a Univalent F'unction 
Rom Hazewinkel [3], page 124, we have: 
Let f(z) be analytic in the unit disk D = { z I IzI < l}, and 
assume that Im f(x) = 0 for -1 < x < 1 and Im f(z) Im z > 0 
for Imz # 0. Then f(z) can be represented as 
where p is a measure concentrated on the circle (5 I 161 = 1) 
and normalized by 11p11 = Jdp(() = 1. 
Pompeiu Formula 
F'rom Henrici [4] we have the following theorem: 
Theorem: Let R be a region bounded by a System I' of 
regular closed curves such that Points in R have winding 
number 1 with respect to I'. If f is a complex-valued function 
that is real-differentiable in a region containing R ü r, then 
for any Point z E R there holds 
where t = x + iy. 
Note that if f (z) is analytic, then this reduces to Cauchy's formula. 
4. F’unction Representation 23 
Solutions to the Biharmonic Equation 
Some function representations require that the function have some 
specific properties. For example, if U is 
(that is, V4u = 0), and if f = V2u, 
z E R): 
biharmonic in a bounded region R 
then u(z) may be written as (for 
1 
dx dy + w(z) 
It - XI 
where t = x + i y and w is harmonic in R (that is, w satisfies Laplace’s 
equation V2w = 0). See Henrici [4]. 
Notes 
[ 11 
[2] 
[3] 
Schläfli’s integral representation is an integral representation of the Legendre 
function of the second kind. 
See also the section on integrals used to represent the solutions ofdifferential 
equations (page 1). 
If a domain D is simply connected, and the vector field V tends suffi- 
ciently rapidly to Zero near the boundary of D and at infinity, then we 
have Helmholtz’s theorem: V = V+ + V x A, where 
+ = - J J J E d v 47rr 
D 
[4] If we define the one-form 
and A = J J J E d V . 47rr 
D 
then a generalization of the Bochner-Martinelli representation, which is 
analogous to the Cauchy-Green formula is given by (See Hazewinkel [3], 
Page 404): 
If the function f is continuously differentiable in the closure 
of the domain D C C” with piecewise-smooth boundary d D , 
then, for any Point z E D , 
24 
Refer ences 
I Applicat ions of Integration 
I. A. Ayzenberg and A. P. Yuzhakov, Integral Representations und Residues 
in Multidimensional Complex Analysis, Translations of Mathematical Mono- 
graphs, Volume 58, Amer. Math. SOC., Providence, Rhode Island, 1983. 
I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, und Products, 
Academic Press, New York, 1980. 
M. Hazewinkel (managing ed.), Encyclopaedia of Mathematics, Kluwer Aca- 
demic Publishers, Dordrecht, The Netherlands, 1988. 
P. Henrici, Applied und Computational Complex Analysis, Volume 3, John 
Wiley & Sons, New York, 1986, pages 290, 302. 
S. Krantz, Function Theory of Several Complex Variables, John Wiley & 
Sons, New York, 1982. 
S. Iyanaga and Y. Kawada, Encyclopedic Dictionary of Mathematics, MIT 
Press, Cambridge, MA, 1980. 
D. Khavinson, “The Cauchy-Green Formula and Its Application to Prob- 
lems in Rational Approximation on Sets with a Finite Perimeter in the 
Complex Plane,’’ J. Funct. Anal., 64, 1985, pages 112-123. 
R. M. Range, Holomorphic Functions und Integral Representations in Sev- 
eral Variables, Springer-Verlag, New York, 1986. 
5. Geornet ric Applicat ions 
Idea 
Integrals and integration have many uses in geometry. 
Length 
U 5 t 5 b, then the length of the curve is given by 
If a two-dimensional curve is parameterized by x(t) = ( z ( t ) , y ( t ) ) for 
For a curve defined by y = y(z), for U 5 z 5 b, this simplifies to 
A curve in three-dimensional space {z( t ) , y ( t ) , ~ ( t ) } , for U 5 t 5 b, has 
lengt h 
5. Geomet ric Applications 25 
Area 
If a surface is described by 
z = f(x, Y ) , for (x, Y ) in the region Rsy, 
then the area of the surface, S, is given by 
If, instead, the surface is described parametrically by x = (x, Y , z ) with 
x = z (u ,v ) , y = y(u,u), z = z(u,u), for (u,u) in the region Ru,,, then the 
area of the surface, S, is given by 
s = J Jlx, x x,,Idudu = d m d u d v , 
Ruv JJ 
R U V 
where 
General Coordinate Systems 
In a three-dimensional orthogonal coordinate system, let { ai} denote 
the unit vectors in each of the three coordinate directions, and let {U,} 
denote distance along each of these axes. The coordinate system may be 
designated by the metric coeficierits (911, g 2 2 , g33}, defined by 
where { X I , x2,23} represent rectangular Coordinates. Then an element of 
area on the u1u2 surface (i.e., u3 is held constant) is given by dS12 = 
[&du11 [&du2]. Metric coefficients for some common orthogonal co- 
Ordinate Systems may be found on Page 113. Moon and Spencer [2] list the 
metric coefficients for 43 different orthogonal coordinate Systems. (These 
consist of 11 general Systems, 21 cylindrical Systems, and 11 rotational 
sys t ems. ) 
Operations for orthogonal coordinate Systems are sometimes written 
in terms of { h i } functions, instead of the {Si,} terms. Here, hi = 6, so 
that, for example, dSl2 = [hldul] [h2du2]. 
26 I Applicat ions of Integration 
Volume 
An element of volume is then given by 
Using the metric coefficients defined in ( 5 . 2 ) , we define g = 911922933. 
Moments of Inertia 
we have t he following definit ions: 
s s p(x, y) dA = M = total mass 
s”, s p(z, y)x dA = My = first moment with respect to the 2-axis ss s p(z, y)y dA = M, = first moment with respect to the y-axis s, s p(x, y)z2 dA = Iy = second moment with respect to the y-axis ss s p(x, y)y2 dA = I, = second momer;t with respect to the 2-axis ss s p ( z , y)(x2 + y2) dA = I0 = polar second moment with respect to the 
Example 1 
[0 ,27r] . The length of this curve is 
For a bounded Set S with positive area A and a density function p(z, y), 
origin. 
Consider a helix defined by x(t) = (U COS t, a sin t, bt) for t in the range 
a2 sin2 t + a2 cos2 t + b2 dt = J”” Ja2+b2 dt = 27rJa2+b2. 
0 
Example 2 
Consider a torus defined by ‘x = ( (b + a sin 4) COS 6, (b + a sin 4) sin 6, 
acos4), where 0 5 6 5 27r and 0 5 4 5 27r. From ( 5 . 1 ) , we can compute 
E = Xe = (b + asin4)2, F = xg - x4 = 0, and G = x4 -x4 = a2. 
Therefore, the surface area of the torus is 
S = 12= 12= d w d 6 d 4 = i2K i2= a(b + a sin 4) d6 d+ = 47r2ab. 
Example 3 
In cylindrical Coordinates we have (21 = r COS 4, 2 2 = r sin 4, 23 = z } 
so that {h, = 1 , he = r , h, = 1 ) . Consider a cylinder of radius R and 
height H . This cylinder has three possible areas we can determine: 
5. Geometric Applications 27 
We can identi@ each of these: Sez is the area of the outside of the cylinder, 
Ser is the area of an end of the cylinder, and Srz is the area of a radial 
slice (that is, a vertical Cross-section fiom the Center of the cylinder). 
We can also compute the volume of this cylinder to be 
hehrhz d8 dr d z = lR 12= r d8 d r d z = x R 2 H . 
= J / x lR 12= 
Notes 
[l] If C is a simple closed curve, positively oriented, that is piecewise continuous, 
then the line integrals jC x d y and - jC y d x both have the Same value, which 
is equal to the area enclosed by C. This is an application of Green’s theorem, 
See page 164. 
The quadratic form (See (5.1)) I = d x - dx = E du2 + 2 F d u d v + G dv2 
is called the first fundamental form of x = x(u,v). The length of a curve 
described by x(u(t) , v ( t ) ) , for t in the range [U, b] is 
[2] 
L = /E ( $)2 + 2 F (2) (2) + G ( z)2 d t . 
[3] The Gauss-Bonnet formula relates the exterior angles of an object with the 
curvature of an object (See Lipschutz [2]): 
Let C be a curvilinear Polygon of class C2 on a patch of a 
surface of class greater than or equal to 3. We presume that 
C has a positive orientation and that its interior on the patch 
is simple connected. Then 
where K g is the geodesic curvature along C , K is the Gaussian 
curvature, R is the union of C and its interior, and the { B i } 
are the exterior angles on C. 
For example, consider a geodesic triangle formed from three geodesics. 
Along a geodesic we have K g = 0, so that Ce Bi = 27r - ss K ds. For a 
planar surface K = 0. Hence, we have found that the sum of the exterior 
angles in a planar triangle is 27r. (This is equivalent to the usual conclusion 
that the sum of the interior angles of a planar triangle is 7r.) 
For a sphere of radius U , we have K = l /a2 . Therefore, the sum of the 
exterior angles on a spherical triangle of area A is 27r - A/u2 . 
R 
28 I Applications of Integration 
Re ferences 
[ 11 
[2] 
[3] 
W. Kaplan, Advanced Calculus, Addison-Wesley Publishing Co., Reading, 
MA, 1952. 
M. M. Lipschutz, Diferential Geornetry, McGraw-Hill Book Company, New 
York, 1969. Schaum Outline Series. 
P. Moon and D. E. Spencer, Fzeld Theory Handbook, Springer-Verlag, New 
York, 1961. 
6. MIT Integration Bee 
Every year at the Massachusetts Institute of Technology (MIT) there 
is an “Integration Bee” Open to undergraduates. This consists of an hour- 
long written exam, with the highest scorers going on to a verbal exam run 
like a Spelling Bee. It is claimed that completion of first Semester calculus 
is adequate to evaluate all of the integrals. 
In 1991 the written exam was given on January 15 and consisted of 
the following forty integrals that had to be evaluated: 
(1) /elgglx dx 
(3) J log x dx 
(5) J.si.2 zecos2 x da: 
(7) /-dx x4 + 1 
(9) /Xex sin x dx 
J 
dx 
sec x + tan 2 sin x 
(13) ) , / z d x 1 + 32 
sin x esec 
dxdx 
(19) /x2 - 1Ox + 26 
dx 
(21) .I 12 + 13cosx 
4 -112 (1 - 4x ) 
dx 
(2) /(sinx - C O S ~ ) ~ dx 
T X 
2 2 + 22 + 2 
(10) /e(.“+.) dx 
(12) /;+‘Pi: dx 
(14) / sinh x - cosh x dx 
(16) J X 4 - 2 2 + 1 dx 
(18) / sec3 x dx 
dx 
(20) /x2 - 112 - 26 
(22) /*dx x + l 
(24) /el”’ dx 
x2 + 1 
6. MIT Integration Bee 29 
(25) /(log x + l)xx dx 
(29) J ~ c s c x - sin x dx 
(31) J42x dx 
(33) Jxex2 dx 
dx 
(35) Jex +e-x 
(37) /cos(sinx) cosxdx 
(39) Jd-. 7r dx 
16 - e2 
(26) /(CO. 2x)(sin6x) dx 
(28) Je'ixx-3 dx 
(32) /z5ex dx 
(36) tan x log I sec X I dx 
(40) 1 J E d x . 
On January 22, the top 11 scorers on the written exam participated in 
the Integration Bee. (These people had obtained between 26 and 35 correct 
answers to the above written exam.) The first few rounds were run with a 
fixed time in which to simplify a specific integral. The integrals, and the 
time allowed for each, were: 
0 1 minute for J sin-' x dx 
x2 - 2x + 2 
x2 + 1 
sin2 x cos2 x 
1 + COS 22 
0 2 minutes for J J.+."J.=dx (since five people in a row did not 
obtain the correct answer, this integral was discarded and the people 
who could not integrate it were not penalized), 
0 2 minutes for J 
0 2 minutes for J 
dx, 
dx, 
0 2 minutes for J cos 4x cos 22 dx, 
dx. 
d~3 - 1 
0 2 minutes for J 
X 
After these integrals, there were four finalists. The ranking of the 
finalists was achieved by four rounds of competitive integration (a pair was 
given the Same integral; whoever obtained the correct answer first was the 
winner of that round). The integrals to be evaluated were 
1 x sec2 x dx 
dx 
xsecx(x tanx+2) dx. 
J log( sin x) 
J 
30 I Applications of Integration 
The 1991 title of “Grand Integrator of MIT” was awarded to Chris 
Teixeira. The second and third place winners were Belle Yseng and Trac 
Tran. 
7. Probability 
Idea 
Procedure 
the expectation of the function g ( X ) is given by 
This section describes how integration is used in probability theory. 
If p ( z ) represents the density function of the random variable X then 
where the range of integration is specified by the density function. Expec- 
tations of certain functions have Special names and notations. For example, 
mean of X = p = p1 = E [ X ] , 
variance of x = Var(X> = u2 = p; = E [ ( X - pl2] , 
n-th moment of X = pn = E [ X n ] , 
n-th central moment of X = p; = E [ ( X - p)”] , 
characteristic function of X = +(t) = E [ e i tx ] , 
generating function of x = ~ ( s ) = E [sx] . 
The random variable X , with density function f(z), has the distri- 
bution function F ( z ) = s_”, f(t) dt. The probability that X < x is then 
given by F ( z ) . 
Notes 
[l] 
[2] 
The mean is sometimes called the “average.” The skewness is defined to be 
to be p3/u3, and the excess is defined to be p4/u4 - 3. 
If the random variable X has the density function f(z), then the entropy of 
X is defined to be (see McEliece [3]) 
H ( X ) = -E[log f ( X ) ] = f(.) log f(z) dz. La 
[3] If the random variable X n (for n = 1 , 2 , . . .) has the distribution Qn (for 
n = 1 ,2 , . . .), respectively, and if 
00 
n+oo lim [I f ( 4 dQn(4 = [, f ( 4 d Q a W 
for every continuous function f with compact Support, then the sequence 
{an} is said to converge in distribution to X,. 
8. Summations: Combinatorial 31 
[4] For a continuous Parameter random variable { X ( t ) } , we can also define 
mean of X = p( t ) = E [ X ] , 
variance of x = Var(t) = E [ ( x ( ~ ) - p( t ) ) ’ ] , 
covariance of X = Cov(s, t ) = E [ ( X ( t ) - p ( t ) ) ( X ( s ) - p ( s ) ) ] , 
References 
[l] 
[2] 
W. Feller, An Introduction to Probability Theory und Its Applications, John 
Wiley & Sons, New York, 1968. 
C. W. Helstrom and J. A. Ritcey, “Evaluation of the Noncentral F-Distri- 
bution by Numerical Contour Integration,” SIAM J. Sci. Stat. Comput., 6, 
No. 3, 1985, pages 505-514. 
R. J. McEliece, The Theory of Information und Coding, Addison-Wesley 
Publishing Co., Reading, MA, 1977. 
[3] 
Surnrnat ions: Cornbinatorial 
Applicable to 
Procedure 
A combinatorial sum may sometimes be written as a Summation over 
contour integrals. Interchanging the Order of integration (when permitted), 
allows a different integral to be evaluated. Evaluating this new integral will 
then yield the desired sum. 
Finding the contour integral representation of the terms in the sum- 
mation rnay be aided by Table 8. 
Example 1 
Evaluation of combinatorial sums. 
Consider the sum 
where m is an integer. By use of Table 8 we make the identification 
If we choose p1 and p2 appropriately (i.e., in this case we require lyI2 > 
4( 1 + x) (1 +Y) in the integration), then we may move the Summation inside 
the integrals and evaluate the sum on k to obtain 
dx dy. (1 + Y>”Y 
xn+’(y2 + 4(1+ y)(l + 2)) 
32 I Applications of Integration 
Table 8. Representations of combinatorial objects i1s contour integrals. Here, 
resF(z) denotes the sum of the residues of F ( z ) at all poles within some region 
centered about the origin. That is: resF(z) = - s, F ( z ) dz. 
X 
1 
X 2Ti 
Binomial Coefficients (where 0 < p < 1): 
Multinomial Coefficients (where r(p) = { x = (21, . . . , Z k ) I 1x1 = pi, 
0 < pa < 1, 2 = 1,. . . , k}): 
~~ 
Bernoulli numbers: B, = n!res (ex - i1-l x-,. 
X 
Euler numbers: E, = n!res cosh-l (z)z-~- ' . 
X 
mn - = res (emxz-n-l) . 
n! x 
Power t erms 
Since rn is an integer, the integral with respect to z may be evaluated by 
the residue theorem to obtain 
Evaluating this last integral, by another application of the residue theorem, 
we obtain our final form for the Summation 
This is one of the so-called Moriety identities. 
8. Summations: Combinatorial 33 
Example 2 
Consider t he summat ion 
where m, n, and p are non-negative integers. Using Table 8, it is easy to 
Show that 
R(m,n,p) := 
The reason that the k Summation can be extended to include large values 
of k is because there are no contributions from these values. By defining 
Sz = {x = ( ~ 1 , ~ 2 , 2 3 ) I 1x11 = 1221 = 2,1x31 = i}, this integral can be 
written as 
If we introduce the new variables tl and t 2 and define the curve St = 
{t = ( t 1 , t z ) I It1l = lt2l = &}, then this last three-dimensionalintegraland 
Summation may be written as t he following five-dimensional contour inte- 
gral: 
dx dt, (1 + 2 3 ) P + l 2 3 R(m, n,p ) = 7 
(2x4 ' I stxs, f(tl,~l>f(~2,~C2>(2122(1+ 2 3 ) - 1)ty+lt;+' 
where f ( a , b ) := 2 3 - a(1 + b ) ( l + 23). If this five-dimensional integral is 
evaluated with respect to 2 1 , 2 2 , and 2 3 , in that Order, then we obtain 
Using Table 8, this two-dimensional integral is equal to 
34 I Applications of Integration 
The final result, equations (8.1) and (8.2)) can be evaluated for differ- 
ent choices of the Parameters to obtain, for instance, 
and 
R(m,m,n) = ( k ) z ( n + ; y ) = (m;n>Z. 
k=O 
Notes 
[l] 
[2] 
Both of the examples in this section are from Egorychev [2], pages 52 and 
169. 
In the paper by Gillis et al. [3] the following representation of the Legendre 
polynomials is used to evaluate the integral J:l Pni ( L ) - - - Pnk (2) dx, where 
n1, . . . , n k are non-negative integers: 
[3] Bressoud [l] uses a combinatorial approach to evaluate integrals of the form 
J::;, - * - J::;, noES sin2’((“) 0 da1 . . . da1, where S is a Set of nonlinear 
sums of elements of the {ai}, and k is an integer-valued function. 
References 
D. M. Bressoud, “Definite Integral Evaluation by Enumeration, Partial Re- 
sults in the MacDonald Conjectures,” Cornbinatoire enumerative, Lecture 
Notes in Mathematics #1234, Springer-Verlag, New York, 1986, pages 48- 
57. 
G. P. Egorychev, Integral Representution und the Cornputation of Cornbina- 
torial Sums, Translations of Mathematical Monographs, 59, Amer. Math. 
SOC., Providence, Rhode Island, 1984. 
J. Gillis, J. Jedwab, and D. Zeilberger, (‘A Combinatorial Interpretation of 
the Integral of the Product of LegendrePolynomials,” SIAM J. Muth. Anal., 
19, No. 6, November 1988, pages 1455-1461. 
K. Mimachi, “A Proof of Ramanujan’s Identity by Use of Loop Integrals,’’ 
SIAM J. Muth. Anal., 19, No. 6, November 1988, pages 1490-1493. 
9. Summations: Other 
Idea 
t egrals. 
Some Summations can be determined by simple manipulations of in- 
9. Summations: Other 35 
c f (m) = & 1 .;rr(cot .;rrz)f(z) dz - Res [.;rr(cot 7rz)f(z)] 
Procedure 
One technique for evaluating infinite Sums is by use of the Watson 
transform (see Page 44). Under suitable convergence and analyticity con- 
straints, we have: 
Theorem: If g(z) is analytic in a domain D with a Jordan 
contour C, then 
. 
& Jc 9 ( z ) cot 7r.z dz = c g(n) 
for those integers n that are within C. 
Alternately (see Iyanaga and Kawada [l], Page 1164): If an analytic func- 
tion f(z) is holomorphic except at poles an (n = 1,2, . . . , k) in a domain 
bounded by the simple closed curve C and containing the Points z = m (for 
m = 1 ,2 , . . . , N ) , then 
(9.1) 
Bzm - BZm (z - LzJ 
f ( 2 " ) ( 4 dz. 
(2m)! 
where t he remainder is R , (n) = 
The remainder can be bounded by 
In this theorem, the Bernoulli polynomials {BS(x)} are defined by the 
generating function 
t s 00 text 
et - i 
-- - CBs(I I : )T . 
S . 
s=o 
The Bernoulli numbers {B,} are given by B, = Bs(0) and a generating 
function for them can be obtained from (9.2), by Setting II: = 0. 
36 
x - - - - 
- N - 1 
I Applicat ions of Integration 
I / 
\ 
\ / C N 
I 
I 
I 
ia * 
I 
I 
- N -4 -3 -2 -1 1 2 3 4 N 
I 
-x- - - - )c +- >t j< - 1- x -x- H -x- - - - x 
-ia A -ia A 
I 
I 
I 
Figure 9.1 Contour for the integral in (9.3). 
Example 1 
S = E,"=, l / (n2 + a2) . We define the integral 
As an example of the Watson transform, consider the sum 
\ 
-> j<- - - 
N + l 
where C N is the contour shown in Figure 9.1. Note that the vertical and 
horizontal sides to CN are at the values -N - 
The contour integral in (9.3) can be evaluated by using Cauchy's 
theorem (see Page 129). The poles within the contour are at z = fia, 
0, f l , f 2 , . . . , f N . The residues at f i a are rcot(fi7ra)/(f2ia), and the 
residue at z = n (for n = 0, f l , . . . , fN) is l /(n2 + a2) . Hence, 
and N + a. 
N r cot ( i r a ) 7r cot ( -i7ra) 
-2ia + + [ 2ia 
n=-N 
As N + 00, it is easy to show from (9.3) that I + 0. Indeed, since 
the cotangent function is bounded, we have I = 0 (IV2) 0 ( N ) + 0 as 
N + 00. Taking the limit as N + 00, and combining (9.3) and (9.4), we 
find 
00 
7r cot (i7ra) r cot ( - i r a ) 
-2ia 
] = o c &+[ 2ia + 
n=-00 
r 1 
= -cothra - -. 1 
00 
2a 2a2 n = l 
(9.5) 
9. Summations: Other 37 
If the limit U + 0 is taken in this formula, then we obtain the well-known 
result (See also Example 3): E,"=, n-2 = 7r2/6. 
Example 2 
the harmonic numbers, defined by Hn = 
find: 
As an example of the Euler-Maclaurin Summation formula, consider 
+ 4 + . . . + ;. Using (9.1) we 
(9.6) 
where 
Taking the limit of n -t 00 in (9.6) results in an expression for Euler's 
constant y: 
00 
B2m - B2m (z - LzJ) dz. Ju ( l+x)2m+1 where the error term is given by E', = 
Example 3 
Zeta function at an argument of two: ((2) := E,"=, n-2. We have 
As an example of a different technique, consider the evaluation of the 
00 M - 
00 4 - 1 
38 I Applicat ions of Integration 
- - 
Since J: x2n dx = i/(2n + i), we can write (J: y2n dy) (J: x~~ dx) = 
1/(2n + l)2. Therefore, we have 
cos U 
sin v sin u 
sin u sin v 
cos v 
cosy cos2 v 
cos2 U COS U 
Now make the Change of variables from {x, Y} to {U, U} via x = sin u/cos 21, 
y = sin vlcos U. The Jacobian of the transformation is given by 
Continuing the calculation of the Zeta function, we find 
C(2) = gl’JI’ dxdy 
1 - x2y2 =fll J ‘ J 2dudv 
0 l - X Y 
(9.7) 
du du. 
The region of integration in the (u,v) plane becomes the triangle with 
vertices at (U = 0,v = 0 ) , (U = 0,v = 7r/2), and (U = 7r/2,u = 0) (see 
Figure 9.2). Since this triangle has area 7r2/8 we finally determine 
47r2 Ir2 C(2) = -- = - 
3 8 6 ’ 
9. Summations: Other 39 
~ ,> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
Figure 9.2 Integration region in (9.7). 
Notes 
[ 11 Under some continuity and convergence assumptions, the Poisson Summa- 
tion formula states (See Iyanaga and Kawada [l], page 924) 
f (n ) = j-i earint f ( t ) d t . 
n=-oa n= - 
(This formula can also be extended to functions of several variables.) 
obtain 
For example, if we take f ( t ) = e-nt2r (for some fixed x > 0), then we 
dt 27rint -rt2 z 
n=-oa n=-oo 
For small values of x, the sum on the right-hand side of (9.8) converges much 
more quickly than the sum on the left-hand side. See also Smith [2]. We 
note in passing that the equatiori in (9.8) represents the following functional 
relationship of theta functions: @(x) = -8 - . 
For another example similar to Example 1, the Summation 
Ja: (3 
[2] 
00 
cosnt - ncosha(7r - t ) 1 - - 
a sinh na 2a2 
n= 1 
7r COS z(n - t ) 
( z 2 + a2) sinnz can be derived from the integral fracl27ri sc d z . 
References 
[l] 
[2] 
S. Iyanaga and Y. Kawada, Encyclopedic Dictionary of Mathematics, MIT 
Press, Cambridge, MA, 1980. 
P. J. Smith, “A New Technique for Calculating Fourier Integrals Based on 
the Poisson Summation Formula,” J. Statist. Comput. Simulation, 33, No. 3, 
1989, pages 135-147. 
R. Wong, Asymptotic Approximation of Integrals, Academic Press, New 
York, 1989. 
[3] 
40 I Applicat ions of Integration 
10. Zeros of Functions 
Applicable to F‘unctions with Zeros that we would like to characterize. 
Idea 
of functions can be obtained. 
By evaluating certain integrals, information about the location of Zeros 
Procedure 
of Zeros of functions. We illustrate two such theorems. 
Redheffer [6], Theorem 6.1): 
There are several theorems that can be used to determine the location 
A Standard theorem from complex analysis states (see Levinson and 
Theorem: Let f(z) be a meroniorphic function in a simply 
connected domain D containing a Jordan contour C. Suppose 
f has no Zeros or poles on C. Let N be the number of Zeros 
and P the number of poles of f in C, where a multiple Zero 
or pole is counted according to its multiplicity. Then 
& e d z = N - P. 
The principle of the argument is the name given to the Statement: 
(10.1) 
(10.2) 
which is just a reforrnulation of (10.1). (Note that “arg” denotes the 
argument or phase of the following function.) The quantity N - P is also 
known as the index of f relative to the contour C. See Example 1 for an 
application of (10.2). 
Another useful theorem is (See Bharucha-Reid and Sambandham [ 13, 
Lemma 4.9): 
Theorem: If f( t) E C1 for a < t < b and f( t ) has a finite 
number of Points in a < t < b with f’(t) = 0, then the 
number of Zeros of f ( t ) in the interval a < t < b is given by 
where multiple Zeros have been counted once. 
See Example 2 for an application of this theorem. 
10. Zeros of Functions 41 
- iRo 0 , R 
I 
Figure 10. The contour used in Example 1. 
Example 1 
Consider the polynomial z3 - z2 + 2 . How many roots does it have 
in the first quadrant? We will use the principle of the argument with 
f(z) = z3 - z2 + 2 and the contour in Figure 10.1 as R + 00. 
To use (10.2), we must determine how the argument of f (z ) changes 
on the three components of the contour C: 
0 The horizontal component (y = 0 and 0 5 x 5 R): We observe that 
f(0) > 0 and f(00) > 0. Since there is only one inflection point of 
f (z) on this component (at z = i), we conclude that f (z ) is always 
positive on this Segment. Hence, there is no Change in argf on this 
Segment : 
argf(z)/ = 0. (10.3) 
0 The curved component z =Reie with 0 5 8 5 5: On this component, 
f (z ) = R3e3ie(l + w) where IwI < 2 / R for large values of R. Hence, 
arg f (Reie) = 38 + arg( 1 + w). Therefore, 
x = R 
x=o 
(10.4) 
I e=o 
where €1 + 0 as R -, 00. 
0 The vertical component (a: = 0 and 0 5 y 5 R): We observe that 
f ( iy) = (-y2 + 2) + i(-y3). As y decreases from R to 0, Ref(iy) 
changes its sign at y = fi from negative to positive, while Imf(iy) 
remains negative. Hence, as y decreases from R to 0, f ( iy) Starts in 
the third quadrant and ends in the fourth quadrant. Therefore, 
37r 7r y=o 
y=R 
(10.5) 
where €2 -+ 0 as R -+ 00. 
Combining the results in (10.3), (10.4), and (10.5) we find that 
As R + 00 we conclude that argf(z)l, = 27r . From (10.2) we conclude, 
therefore, that there is exactly one root of f (z ) in the first quadrant. 
42 I Applications of Integration 
Example 2 
In this example we answer the question: “What is the average number 
of real roots of the polynomial f(x; a) := ao + u1x + . . . + an-lxn-’ when 
the {a i } are Chosen randomly?” The analysis here is from Kac [5] ; See also 
Bharucha-Reid and Sambandham [l]. 
For definiteness, we presume that the {ui} lie on the surface of the 
n-dimensional sphere of radius unity, Sn(1) (i.e., the {ui} satisfy laI2 = 
E:=,’ U: = 1). Define Nn(a) to be the number of real roots of f(x; a). 
A simple scaling argument Shows that Nn(aa) = Nn(a) for any non-Zero 
constant a; this will be needed later. 
Define Mn to be the average number of real roots of the polynomial 
f(x;a) as a varies over Sn(1). That is, 
(Here du represents the surface element on a sphere.) It is not difficult to 
Show that Mn can also be represented in the form 
(where da = da0 da1 . . . dan-l), since this last integral can be rewritten as 
follows: 
If we we use the notation ()(l) to denote the number (or fraction) of 
real roots in the interval ( - 1 , l ) and ()(2) to denote the number (or fraction) 
of real roots not in the interval (-l,l), then Ni’)(a) = Ni2)(a) because 
~~~~ f&xk = (x:~: an-l-kx-k) xn-1. (That is, for every polynomial 
having x as a root, there is a corresponding polynomial with x-’ as a 
root.) This implies that M?)(a) = Mp)(a) and Mn(a) = 2M?)(a). 
Using the second theorem stated in the procedure, with U = -1 and 
b = 1, we find that 
. I 
(10.6) 
10. Zeros of Functions 43 
(Note that we write f ( t ) for the function f ( t ; a).) Interchanging the Order 
of integration, and recognizing that the absolute value function has the 
simple integral representation Iyl = - d q , allows (10.6) to 
be written as 
lr Jrn -00 l-cOsqy v2 
where 
and 
Writing the cosine function in complex exponential form allows the a- 
integrals to be evaluated (recall that f ( t ) = f ( t ; a)) to determine that 
Hence, 
If we define the functions An(t) = 
Cn(t ) = 
t k , Bn(t) = kt2"', and 
k2 t2k , then we can finally find 
Our final answer is therefore 
d t . (10.7) l 1 Mn = 2M:) = 1, 1 - t2 
2 
An asymptotic analysis then reveals that Mn M - log n as n + 00. 
lr 
44 
Notes 
I Applicat ions of Integration 
The Same result in (10.7) is obtained in three different cases: 
(A) The {a i } are Chosen to be uniformly distributed on the interval (-1,l); 
(B) The {a i } are Chosen to be equal to +1 and -1 with equal probability; 
(C) The {ai} are Chosen to be uniformly distributed on the unit ball (as 
If C* is the image of C under f in the first theorem in the Procedure, then 
N - P turns out to be equal to the number of times C* Winds around the 
origin; i.e., it is the winding number of C* with respect to the origin. 
In example 1, the polynomial f(z) = z3 - z2 + 2 has the roots z = -1 and 
z = l f i . 
If g ( z ) is analytic in D , the zeros of f(z) are simple and occur at the Points 
{a i } , the poles of f(z) are simple and occur at the Points { b i } , then the 
result in (10.1) may be extended to 
shown in Example 2). 
(10.8) 
If we choose f ( z ) = sin r z , then (10.8) becomes 
for those integers n that are within the contour C. This formula is very 
useful for evaluating infinite Sums; it is known as the Watson transform. An 
example of its usage may be found in the section beginning on page 34. 
Rouche’s theorem compares the number of zeros of two related functions. It 
states (See Levinson and Redheffer [6], Theorem 6.2): 
Let f(z) and g ( z ) be analytic in a simple connected domain 
D containing a Jordan contour C. Let If(z)l > [ g ( z ) \ on C. 
Then f(z) and f(z) + g ( z ) have the Same number of zeros 
inside C. 
This theorem can be used to prove that a polynomial of degree n has n roots. 
For the polynomial h ( z ) = aizi of degree n, choose f ( z ) = aozn and 
9 w = h ( z ) - f ( 4 . 
References 
[l] 
[2] 
A. T. Bharucha-Reid and M. Sambandham, Random Polynomials, Academic 
Press, New York, 1986, Chapter 4, pages 49-102. 
M. P. Carpentier, “Computation of the Index of an Analytic Function,” in 
0. Keast and G. Fairweather (eds.), Nurnerical Integration: Recent Dewel- 
opments, Software und Applications, Reidel, Dordrecht , The Netherlands, 
1987, pages 83-90. 
M. P. Carpentier and A. F. DOS Santos, “Solution of Equations Involving 
Analytic Functions,” J. Comput. Physics, 45, 1982, pages 210-220. 
[3] 
11. Miscellaneous Applications 45 
[4] N. I. Ioakimidis, “Quadrature Methods for the Determination of Zeros of 
Transcendental Functions-a Review,’’ in P. Keast and G . Fairweather (eds.), 
Nurnerical Integration: Recent Developments, Software and Applications, 
Reidel, Dordrecht, The Netherlands, 1987, pages 61-82. 
M. Kac, “On the Average Number of Real Roots of a Random Algebraic 
Equation,” Bu1E. Amer. Math. SOC., 49, pages 314-320, 1943. 
N. Levinson and R. M. Redheffer, Complex Variables, Holden-Day, Inc., San 
F’rancisco, 1970, Section 4.6, pages 216-223. 
[5] 
[6] 
11. Miscellaneous Applicat ions 
Idea 
This section describes other uses of integration. 
P hysics 
Let F be the force on an object in three-dimensional space. The work 
done in moving an object from Point a to Point b is defined by the line 
integral 
b 
W = l F - d s , 
where s is an element of the path traversed from a to b. In a conservative 
force field, the force can be written as the gradient of a scalar potential 
field: F = VP. In this case, the amount of work performed is independent 
of the path and is given by W = P(b) - P(a). 
For example, since gravity is a conservative force field, g = (0, 0, g ) = 
V ( g z ) , the work performed in moviag an object from the location a = 
(a„a„a,) to the location b = ( b z , b„b,) is just W = g(bz - U , ) . 
Mechanics 
The momentum of a rigid body K is defined by Q = J(dr/dt) dm 
where dm is the mass of the volume element at a point r and ( d r l d t ) is its 
velocity. See Iyanaga and Kawada [6], page 454. 
The angular momentum of a rigid body, about an arbitrary Point ro, 
is defined by H = s ( r - ro) x ( d r l d t ) dm. See Iyanaga and Kawada [6], 
page 455. 
Mechanics 
If V2u = -1 in G, and U = 0 along dG, then the torsional r igidi ty of 
the domain G is defined to be P = 4 J’ U dA. For a disk of radius R, we 
have P = nR4/2. See Hersch [4]. 
G 
46 I Applicat ions of Integration 
Notes 
[l] 
[2] 
Using an integral representation of derivatives, Calio et ul. [2] demonstrate 
how quadrature formulas can be used to differentiate analytic functions. 
Let Q be a bounded two-dimensional domain with a partly smooth curve of 
boundary. Aizenberg [ 11 contains an integral representation for the difference 
between the number of lattice Points of Q and its volume. Then a similar 
result for a three-dimensional domain is given. 
Ioakimidis [5] uses contour integrals to find the location of branch Points. [3] 
References 
[l] L. A. Aizenberg, “Application of the Multidimensional Logarithmic Residue 
to Number Theory. An Integral Formula for the Difference Between the 
Number of Lattice Points in a Domain and its Volume,” Ann. Polon. Muth.,46, 1985, pages 395-401. 
F. Calio, M. F’rontini, and G. V. MilovanoviC., “Numerical Differentiation of 
Analytic Functions Using Quadratures on the Semicircle,” Comp. & Muths. 
with Appls., 22 , No. 10, 1991, pages 99-106. 
H. M. Haitjema, “Evaluating Solid Angles Using Contour Integrals,” Appl. 
Math. Modelling 11, No. 1, 1987, pages 69-71. 
J. Hersch, “Isoperimetric Monotonicity: Some Properties and Conjectures 
(Connections Between Isoperimetric Inequalities) ,)’ SIAM Review, 30, No. 4, 
December 1988, pages 551-577. 
N. I. Ioakimidis, “Locating Branch Points of Sectionally Analytic Functions 
by Using Contour Integrals and Numerical Integration Rules,” Int. J. Comp. 
Muth., 41, 1992, pages 215-222. 
S. Iyanaga and Y. Kawada, Encyclopedic Dictionury of Muthemutics, MIT 
Press, Cambridge, MA, 1980. 
I. Vardi, “Integrals, an Introduction to Analytic Number Theory,” Amer. 
Muth. Monthly, 95, No. 4, 1988, pages 308-315. 
[2] 
131 
[4] 
[5] 
[6] 
[7] 
I1 
Concepts and 
Definitions 
12. Definitions 
Asymptotic Expansion Let f(x) be continuous in a region R and let 
{ + n ( x ) } be an asymptotic sequence as x .-) 20. Then the formal series 
C,"=,an&(x) is said to be an infinite asymptotic expansion of f(z), as 
z + $0, with respect to ( 4 n ( x ) } if the equivalent Sets of conditions 
m 
f(z> = C an$n(x:> + O ( 4 m + i ( x > > , 8s x -, zo, (12.1) 
n=O 
r m 1 
n=O 1 4m(z) J = O , (12.2) 
for each m = 0,1,2,. . . are satisfied. If, instead, (12.1) only holds for 
m = 0,1,2 ,..., N - 1, then 
N - 1 
n=O 
47 
48 I1 Concepts and Definitions 
F, Fv F, ... 
- H , Hv H , ... 
a(F,G,H, ...) - GU Gv G w . . * 
a(u, U , w, . . .) - 
When used in a Change of variable computation (See page log), the absolute 
value of the Jacobian is used. 
12. Definitions 49 
L Functions A measurable function f(x) is said to belong to Lp if JF If(x)lPdx is finite. 
Lebesgue Measurable Set A set of RP is called a Lebesgue measurable 
set (or simply a measurable set) if it belongs to the smallest a-algebra 
containing the Bore1 sets and the sets of measure Zero of RP. 
Leibniz’s Rule Leibniz’s rule states that 
Linear Independence 
Wronskian is t he det erminant 
Given the smooth functions {yl, y2,. . . , yn}, the 
If the Wronskian does not vanish in an interval, then the functions are 
linearly independent . 
Lower Limit, Upper Limit In the integral s,” f(x) dx, the Zower Zimit 
is the value U, the upper Zimit is the value b. 
Measure A (positive) measure on a a-algebra A is a mapping p of A 
into [0, m] such that if E is a disjoint Union of a sequence of sets E n E A, 
Meromorphic Function A miromorphic function is analytic, except 
possibly for the presence of poles. 
Norms 
then P(E) = p(En). 
If f is a measurable function on Rn then we define the L, norm 
Orthogonal Two functions f(z) and g ( x ) are said to be orthogonal 
with respect to a weighting function w ( x ) if the inner product vanishes, 
i.e., (f(x), g ( x ) ) := f ( x ) w ( x ) g ( x ) dx = 0 over some appropriate range of 
integration. Here, an overbar indicates the complex conjugate. 
Pole An isolated singularity of f (z ) at a is said to be a pole if f (z ) = 
g(x ) / ( z -a ! )” , where m 2 1 is an integer, g ( z ) is analytic in a neighborhood 
if a! and g ( a ) # 0. The integer rn is called the Order of the pole. 
Sigma-algebra A family A of subsets of a set X is called a a-algebra 
if the empty set is in A, and if A is closed under complementation and 
countable Union. 
50 I1 Concepts and Definitions 
Set of Measure Zero A Set E in R p is called a Set of measure Zero (or 
a negligible Set) if there exists a Bore1 Set A such that E c A and v ( A ) = 0. 
Variations Let f(x) be a real bounded function defined on [U, b]. Given 
the subdivision U = xo < x1 < . . . < xn = b, denote the sum of positive 
(negative) differences f(xi) - f(xi-1) by P ( - N ) . The suprema of P , N , 
and P + N , for all possible subdivisions of [U, b] , are called the positive 
Variation, the negative Variation, and the total variation of f(x) on [U, b]. 
If any one of these three values is finite, then they are all finite. In this 
case, f (x) is said to be of bounded Variation. 
The continuous function xsin $ is not of bounded Variation, while the 
discontinuous function sgn z is of bounded Variation. 
If g(z) is an increasing function on [u,b] then the total Variation of 
g(x) on [u,b], written ~ a r [ ~ , b ~ g , is given by g(b) - g(u). Hence, by writing 
an arbitrary continuous function f(x) as the difference of two increasing 
functions f(x) = fl(x) - f2(x) we find 
var[a,b]f = Var[a,b]fl + Var[a,b]fl = ( f l ( b ) - f l (4) + ( f 2 ( b ) - f 2 ( 4 ) . 
One way to form the decomposition f(x) = fl(z) - f2(z) is by 
(Note that the notation [ I - and [ I f is defined on Page 352.) 
Bounded Variation A function g(x) is of bounded Variation in [U, b] if 
and only if there exists a number M such that E:, Ig(xt.i) -g(zi-l)l < M 
for all partitions U = xo < x1 < 2 2 < ... < x, = b of the interval. 
Alternately, g ( x ) is of bounded Variation if and only if it can be written 
in the form g(x) = gl(x) - 92(x) where the functions gi(x) and 92(x) are 
bounded and nondecreasing in [U, b]. 
Weyl’s Integral Formula Let G be a compact connected semisimple 
Lie group and H a Cartan subgroup of G. If p, ß, and X are all normalized 
to be of total measure 1, then 
for every continuous function f on G, where w is the Order of the Weyl 
group of G. Here J is given by 
where P is the Set of all positive roots QI of G with respect to H and X is 
an arbitrary element of the Lie algebra of H . 
13. Integral Definitions 51 
13. Integral Definitions 
Idea 
include t he following: 
There are many different types of integrals of interest. These integrals 
Abelian (see below) 
contour (see Page 129) 
fractional (see Page 75) 
improper (see below) 
Lebesgue (see below) 
loop (see Page 4) 
Riemann (see below) 
Stratonovich (see Page 186) 
Cauchy (see Page 92) 
Feynman (see Page 70) 
Henstock (see below) 
Ito (see Page 186) 
line (see Page 164) 
path (see Page 86) 
stochastic (see below) 
surface (see Page 24) 
Properties of Integrals 
function should have. These properties are: 
Lebesgue [16] defined six properties that the integral of a bounded 
Squire [22] indicates that more than sixty kinds of integrals have been 
developed that satisSl the above criteria, in different degrees of generality. 
In the following sections we describe only a few of the different types of 
integrals. Pesin [20] has a very comprehensive review of many types of 
integrals. 
Abelian Integral 
Suppose that we have an algebraic curve whose equation is G(z, y) = 0. 
Let y = f(z) be the algebraic function satisfying this equation, and define 
S to be the associated Riemann surface on which y is Single-valued. Define 
the rational function R by R(z, y) = P(z , y)/Q(z, y), where both P and Q 
are polynomial functions. Note that R is Single-valued on S and that the 
only singularities that R has on S are a finite number of poles. 
An Abelian integral has the form I ( z , y) = ~ z ~ ~ , R(z, y) dx, where the 
path of integration is on the surface S. The value of this integral depends 
upon the integration path. Note that I ( z , y ) is regular for all finite paths 
52 I1 Concepts and Definitions 
f ( t 2 ) (22 - X i - 1 ) - I 
i= 1 
that avoid the poles of the integrand. There are only three kinds of Abelian 
integrals; an Abelian integral is of 
[l] the first kind if it is regular everywhere, 
[2] the second kind if its only singularities are poles, 
[3] the third kind if it has logarithmic singularities. 
No other types of singularities are possible for an Abelian integral. 
Note that an Abelian integral can be of the first kind and not be 
constant; Liouville's theorem does not apply since I is defined on a Ftiemann 
surface, not the complex plane. 
If the limits of integration are fixed, then all possiblevalues of an 
Abelian integral can be determined by considering t he combinatorial topol- 
ogy of S. Fixing the Points A and B on S, define J = Jf R ( x , y) dx. If P is 
a specific path on S from A to B, then any other path from A to B is of the 
form P + r, where r is a closed path passing through A and B. Define K 
to be the Abelian integral associated with the path r: K = Jr R(x , y) dx. 
Note that the value of K is not changed as r is continuously distorted, 
provided that I' stays on S and does not Cross any poles of R(x,y). 
As an example, elliptic integrals can be defined by: R(x , y) = l / y and 
Much research has been performed on the inversion of Abelian inte- 
G(x , Y) = (1 - x2)( i - k2x2) - y2 = 0. 
grals. For example, Theorem 6.2 of Bliss [l] (page 170) states that: 
Theorem: If an Abelian integral U = s((O~ol q(z, y) dz on 
the Riemann surface T of an irreducible algebraic equation 
f(z, y) = 0 defines a Single-valued inverse function z(u), y(u), 
then the genus of the curve f = 0 must be either p = 0 or 
p = 1. In the case p = 0 the integral is either of the second 
kind with a Single simple pole, or of the third kind with two 
simple logarithmic places and no other singularities. In the 
case p = 1 the integral is of the first kind. 
For details, see Hazewinkel [8] (pages 14-16), or Lang [15]. 
< E. 
Henstock Integral 
Given the interval [U, b] and a positive function S : [U, b] + R, define a 
Partition to be given by {(ti , [~i-l,xi])};=~, where the intervals [xi-l,xi] 
are non-overlapping, their Union is the interval [U, b], and the following con- 
dition is satisfied: ti E [xi-l, xi] c (ti - 6( t , ) , ti + S ( t i ) ) , for i = 1 , 2 , . . . , n. 
A function f : [u,b] -+ R is called Henstock-integrable if there exists 
a number I such that for every E > 0 there exists a positive function 
S : [U, b] + R such that every Partition of the interval [U, b] results in 
I n I 
13. Integral Definitions 53 
The number I , usually written as s,” f(t) d t , is called the Henstock integral 
of f . For details, see Peng-Yee [19]. 
Improper Integrals 
An integral in which the integrand is not bounded, or the interval of 
integration is unbounded, is said to be an improper integral. For example, 
the following are improper integrals: 
Suppose that f(x) has the Singular point z in the interval (U, b ) , and 
suppose that f(x) is integrable everywhere in the interval, except at the 
point x. The integral s,” f(x) d x is then defined to have the value 
where the limits are to be evaluated independently. 
Lebesgue Integral 
Let X be a space with a non-negative complete countably-additive 
measure p, where p ( X ) < 00. Separate X into { X n } so that U:=iXn = X . 
A simple function g is a measurable function that takes at most a countable 
set of values; that is g ( x ) = Y n , with Y n # yk for n # k, if 2 E Xn. A simple 
function g is said to be summable if the series E,”=, YnpX, converges 
absolutely; the sum of this series is the Lebesgue integral sx g d p . 
A function f : X + R is summable on X (denoted f E & ( X , p ) ) if 
there is a sequence of simple, summable functions { g n } , uniformly conver- 
gent to f on a set of full measure, and if the limit limn-,, sx gn d p is finite. 
The number I is the Lebesgue integral of the function f; this is written 
I = J x f d c . (13.1) 
A simple figure can clarify how the Lebesgue integral is evaluated. 
Given the function f(x) on (u ,b) , subdivide the vertical axis into n + 1 
Points: minf 2 yo < y1 < . - . < Y n 2 maxf. Then form the sum 
n 
i=l 
in which the measure is the sum of the lengths of the subintervals on 
which the stated inequality takes place. (See Figure 13.1.) In the limit of 
n + 00, as the largest length (yi - yi-1) tends to Zero, this sum becomes 
the Lebesgue integral of f(x) from U to b. 
The Lebesgue integral is a linear non-negative functional with the 
following properties: 
54 I1 Concepts and Definitions 
Figure 
Regions 
13.1 A schematic of how the Lebesgue integral is to be eval 
with similar values are shaded in the Same way. 
Y5 
Y1 
a - 
a b x ’ 
13.1 A schematic of how the Lebesgue integral is to be eval 
with similar values are shaded in the Same way. 
.uated. 
[l] If f E L1(X, p ) and if p {x E X I f (x ) # h(x)} = 0, then h E L1(X, p ) 
and Jx f d p = sx hdp. 
[3] If f E Ll(X,p), lhl I f and h is measurable, then h E Ll(X,p) and 
[4] If m 5 f I M and if f is measurable, then f E L l ( X , p ) and m pX I 
[21 If f E Ll(X,P), then lfl E Ll(X,P) arid IJX fdPl I Jx Ifl dP. 
IJX h dPl I Jx f dP* 
J x f d P I M P X . 
For functions from Rn to R, if the measure used is the Lebesgue 
measure, then (13.1) may be written as J = JRn f(x) dx. For different 
measures, the functional J is called a Lebesgue-Stieltjes integral. 
Every function that is Riemann integrable on a bounded interval is 
also Lebesgue integrable, but the converse is not true. For example, the 
Dirichlet function (equal to 0 for irrational arguments, but equal to 1 for 
rational arguments) is Lebesgue integrable but not Riemann integrable. 
Alternatively, the existence of an improper Riemann integral does not 
imply the existence of a Lebesgue integral. For example, the integral Jr sinx/x dx = 7r/2 is not Lebesgue integrable because Lebesgue inte- 
grability requires that both f and l f l should be integrable. In this example 
Jom I sinxl/xdx = 00. 
Riemann Integral 
Let f (x) be a bounded real-value function defined on the interval I = 
[U, b]. Denote a Partition of I by D = {xo,. . . ,x,} where a = xo < 21 < 
. . . < x, = b and n is finite. Let Ii denote the sub-interval [xi,xi+l]. Define 
13. Integral Definitions 55 
Figure 13.2 An illustration of the lower (left) and upper (right) Sums of a 
function. 
The oscillation of f on Ii is defined to be Mi -mi. Now define the Darboux 
Sums a ( D ) and g(D) : 
n n 
a ( D ) = C M z ( X i - X i - l ) , g ( D ) = ( X i - X i - 1 ) - 
i=l i=l 
Considering all possible partitions of D, we define 
Riemann upper integral of f = f ( x ) dx = inf a ( D ) , 
D 
b 
Riemann lower integral of f = f(x) dx = supg(D). 1 D 
(See Figure 13.2.) If the Riemann upper and lower integrals of f coincide, 
then the common value is called the Riemann integral of f on [U, b] and is 
denoted by S,bf(x)dx. In this case, the function f is said to be R i e m a n n 
integrable, or just integrable. 
Darboux’s theorem states that: 
Theorem (Dwbouz): For each E > 0 there exists a positive 6 
such that the inequalities 
hold for any Partition D with max(zi - zi+~) < 6, for i = 
1 , 2 ,..., n. 
From Darboux’s theorem we conclude that necessary and sufficient 
conditions for a function f ( x ) to be integrable on [u,b] is that for each 
positive E there exists a 6 such that 
where 6(D) = maxi(xi - xi-1) < S and Ci is Chosen arbitrarily from Ii. 
56 I1 Concepts and Definitions 
Stochastic Integrals 
Let X(t) be an arbitrary random process defined in some interval 
U 5 t 5 b and let h ( t , ~ ) be an arbitrary deterministic function defined 
in the Same interval. Define the integral I(.) = Ja h(t , .)X(t) dt . If the 
integral exists, then I(.) is, itself, a random variable. 
b 
The integral can be shown to exist for each sample function x ( t ) if 
Furthermore, when the integral exists, we can write 
and the interchange of the Order of integration and the expectation opera- 
tion is justified. 
Even if the integral does not 
function x ( t ) of X(t), it may be 
stochastic sense. (See page 186.) 
Notes 
Other properties of an integral 
exist in the usual sense for each sample 
possible to define the equality in some 
can be inferred from the stated properties 
of integrals. Let I denote the Set of all functions integrable on the interval 
I = [a ,b] . If f and g belong to I, and a and p are arbitrary real numbers 
t hen 
(Al Ifl E 1, 
(B) a f+ß9 E 1, 
(C ) f * 9 a 
(D) min{f,9) E 1, 
(E) max{f,9) EI, 
(F) f /g E I (assuming that 191 2 A > 0 on I ) . 
We also have the conventions Jaa f(x) dx = 0 and Jba f(x) dx = - Ja f(x) dx. 
Botsko [3] describes a generalization of the Riemann integral that admits 
every derivative into the Set of integrable functions. As an example, Botsko 
considers the function f‘ where 
b 
if x = 0. 
The function f’ has an unbounded derivative and is not Lebesgue integrable, 
but can be integrated with Botsko’s integral. 
The Burkill integral [8] was originally introduced to determine surface areas, 
See Burkill [4]. In modern usage, it is used for integration of non-additive 
functions. The Burkill integral is less general than the subsequently intro- 
duced Kolmogorov integral; any function that is Burkill-integrable is also 
Kolmogorov-integrable. The name of “Burkill integral’’ is also given to a 
number of generalizations of the Perron integral [20]. 
Integral Definitions 57 
The Henstock integral is also known as the generalized Riemann integral. 
The Henstock integral and the Denjoy integral are equivalent. The restricted 
Denjoy integral includes the Newton integral and the Lebesgue integral. 
The Perron integral, the Luzin integral, the gauge integral, the Kurzweil- 
Henstock integral, and the Special Denjoy integral are all equivalent. See 
Henstock [9]. In one dimension, the Perron integral is equivalent to the 
restricted Denjoy integral. A multi-dimensional Perron integral is described 
in Jurkat and Knizia [13]. 
The Boks integral [8] is a generalization of the Lebesgue integral, first 
proposed by Denjoy, but studied in detail by Boks [2]. The definition Starts 
by taking a real-valued function f defined on a Segment [U, b] and periodically 
extending it to the entire real line (with period b - U). The A-integral [8] is 
more convenient to use than the Boks integral. 
Other types of integrals not described in this book include: 
(A) Banach integrals, Birkhoff integrals, Bochner integrals, Denjoy inte- 
grals, Dunford integrals, Gel’fand-Pettic integrals, and harmonic inte- 
grals (See Iyanaga and Kawada [12], pages 12-15, 337-340, 627-629, 
787). 
(B) Norm integrals, refinement integrals, gauge integrals, Perron integrals, 
absolute integrals, general Denjoy integrals, and strong variational in- 
tegrals (See Henstock [9]). 
(C) Borel’s integral, Daniell’s integral, Denjoy integral, improper Dirichlet 
integrals, Harnack integrals, Hölder’s integrals, Khinchin’s integrals, 
Radon’s integrals, Young’s integrals, and De la Vallee-Possin’s integrals 
(See Pesin [2O]). 
(D) Curvilinear integrals are better known as line integrals, See page 164. 
(E) Fuzzy integrals (See Ichihashi et ul. [ l l ] ) . 
(F) Kolmogorov integrals (See Hazewinkel [8], page 296). 
It is also possible to define an integral over an algebraic structure. For 
example, integrals in a Grassman algebra are discussed in de Souza and 
Thomas [21]. 
The Lommel integrals are specific analytical formulas for the integration of 
products of Bessel functions. See Iyanaga and Kawada [12], page 155. 
The notion of stochastic integration was first introduced by Wiener in con- 
nection with his studies of the Brownian motion process. Given a one- 
dimensional path X ( t ) and any function f ( t ) , Wiener wanted to be able to 
define the integral J: f( t ) d X ( t ) . This integral makes no sense as a Stieltjes 
sum since X ( t ) is not a function of bounded Variation. 
References 
[l] 
[2] 
[3] 
[4] 
G. A. Bliss, Algebraic Functions, Dover Publications, Inc., New York, 1966. 
T. J. Boks, “Sur les rapports entre les methodes de l’integration de Riemann 
et de Lebesgue,” Rend. Circ. Mut. Palermo, 45, No. 2, 1921, pages 211-264. 
M. W. Botsko, “An Easy Generalization of the Riemann Integral,” Amer. 
Muth. Monthly, 93, No. 9, November 1986, pages 728-732. 
J . C. Burkill, “Functions of Intervals,” Proc. London. Math. Soc., 22, No. 2, 
1924, pages 275-310. 
I1 Concepts and Definitions 
G. F. Carrier, M. Krook, and C. E. Pearson, Functions of a Complex Vari- 
able, McGraw-Hill Book Company, New York, 1966. 
P. J . Daniell, “A General Form of Integral,” Ann. of Math., 19, 1918, pages 
A. Denjoy, ((Une extension de l’integrale de M. Lebesgue,” C. R. Acad. Sci., 
154, 1912, pages 859-862. 
M. Hazewinkel (managing ed.), Encyclopaedia of Mathematics, Kluwer Aca- 
demic Publishers, Dordrecht, The Netherlands, 1988. 
R. Henstock, The General Theory of Integration, Oxford University Press, 
New York, 1991. 
W. V. D. Hodge, The Theory und Application of Harmonic Integrals, Cam- 
bridge University Press, New York, 1989. 
H. Ichihashi, H. Tanaka, and K. Asai, “Fuzzy Integrals Based on Pseudo- 
Additions and Multiplications,” J. Math. Anal. Appl., 130, 1988, pages 354- 
364. 
S. Iyanaga and Y. Kawada, Encyclopedic Dictionary of Mathematics, MIT 
Press, Cambridge, MA, 1980. 
W. B. Jurkat and R. W. Knizia, “A Characterization of Multi-Dimensional 
Perron Integrals and the Fundamental Theorem,’’ Can. J. Math., 43, No. 3, 
1991, pages 526-539. 
A. Ya. Khinchin, “Sur une extension de l’integrale de M. Denjoy,” C. R. 
Acad. Sci., 162, 1916, pages 287-291. 
S. Lang, Introduction to Algebraic und Abelian Functions, Addison-Wesley 
Publishing Co., Reading, MA, 1971. 
M. Lebesgue, “Leqons sur l’integration,” Gauthier-Villars, Paris, Second 
Edition, 1928, page 105. 
E. J. McShane, “Integrals Devised for Special Purposes,” Bull. Amer. Math. 
SOC., No. 5, September 1963, pages 597-627. 
R. M. McLeod, The Generalized Riemann Integral, Mathematical Associa- 
tion of America, Providence, RI, 1980. 
L. Peng-Yee, “Lanzhou Lectures on Henstock Integration,” World Scientific, 
Singapore, 1989. 
I. N. Pesin, “Classical and Modern Integration Theories,” translated by S. 
Kotz, Academic Press, New York, 1970. 
S. M. de Souza and M. T. Thomas, “Beyond Gaussian Integrals in Grassman 
Algebra,” J. Math. Physics, 31, No. 6, June 1990, pages 1297-1299. 
W. Squire, Integration fo r Engineers und Scientists, American Elsevier Pub- 
lishing Company, New York, 1970. 
279-294. 
14. Caveats 
Idea 
rect. 
There are many ways in which an integration “result” may be incor- 
14. Caveats 59 
Example 1 
Consider the integral 
I (u ,b) = Jol 
ux + b’ (14.1) 
for real x and arbitrary nonzero complex U and b. The indefinite integral 
has the primitive log(uz + b)/u. Hence, a careless “direct” derivation would 
yield t he result 
? log(u+b) logb I (u ,b) = -- 
U U 
(14.2) 
The Problem, of Course, is that the logarithm function has a branch 
Cut. Hence, the two logarithms in (14.2) may not be on the Same Riemann 
sheet. 
The correct way to evaluate (14.1) is to separate the region of integra- 
tion into two sub-intervals, with the division Point being the value where 
ux + b may vanish. An easier way, for this integral, is to first write the 
integral as 
No matter what the sign is of Im(b/u), the argument of the logarithm never 
crosses the Cut (since x is real). Thus, the answer is 
I (u ,b) = - log 1 + - - log- . 
U [ ( 3 :I 
Note that since 1 + b/u and b/u have the same imaginary part, we may 
combine them to obtain our final answer 
1 u + b 
U b . 
I(U) b) = - log - 
Example 2 
Consider t he simple integral 
1: 2 logx. (14.3) 
Clearly, the integrand (l /x) is an odd function, yet logx is neither an even 
nor an odd function. Hence, there must be an error in (14.3). The error in 
this case is simple, the integral should be written as 
J $ = log 1x1. 
60 I1 Concepts and Definitions 
dx 2 tan- 2 + 1 
J3 
Now the fact that the result is an even function is clearly indicated. 
As a similar example, consider the integral 
0 (14.5) 
dx ? X I = J dm = sin-1 -. 
U 
(14.4) 
when 1x1 5 U. The integrand is an even function of both x and U, so that 
the formula for the integral should be odd in x and even in U. However, 
the formula in (14.4) is odd with respect to both x and U. The correct 
evaluation of the integral in (14.4) can be written as I = sin-l(x/lul) or as 
I = tan-l 
Example3 
an elementary indefinite integral we readily find 
X 
Jm' 
Consider the integral I = so" dx/f2 + sinx). Since this integrand has 
Now we must carefully consider what branch of the inverse tangent function 
to take in each of these two terms. In this case, we must take 
41r 
so that I = - [ ( 2 l r + 3 - (31 = J3. J3 
Notes 
[l] 
[2] 
Very few tables of integrals use absolute value signs, such as are required in 
Example 2. 
If the principle branch of the inverse tangent function were Chosen for both 
terms in (14.5), then the calculation would return the value Zero. This 
should be identified immediately as being in error, since the integrand is 
always positive. 
Observe that contour integration techniques (See page 129) can quickly 
yield the correct evaluation of I in Example 3. 
Particular care must be used when a symbolic manipulation package (See 
page 117) is used to compute a definite integral. Results from such packages 
may be incorrect for many reasons; choosing the wrong branch Cut, as 
Example 3 illustrates, is a common error. 
[3] 
15. Changing Order of Integration 61 
15. Changing Order of Integration 
Applicable to Multiple integrals. 
Yields 
A different multiple integral. 
Idea 
This may make it easier to evaluate the integral. 
Sometimes the Order of integration in a multiple integral can be changed. 
Procedure 
tiple integral may be changed. Fubini's theorem states: 
When Fubini's theorem is satisfied, the Order of integration in a mul- 
Theorem: (Fubini) If f ( x , Y) is measurable and non-negative, 
and any one of the integrals 
exists, then the other integrals exist, and all are equal. 
For principal-value integrals, t he Poincare-Bertrand t heorem may be 
useful (See Muskelishvili [3]): 
Theorem: (Poincur&-Bertrand) If f ( x , Y) is analytic, then 
This theorem is also true with weaker conditions on f(x). 
62 I1 Concepts and Definitions 
Example 
Consider the double integral 
I=Jol J' dxdy. 
0 (xy2 + 1) 
The integrand is positive in the region of integration. In the process of 
evaluating the integral we will determine if it exists. Integrating first with 
respect to x we have 
Y 1 1l d y l dx (xy2 + 1)2 
Jol dy ( Y (x,:+ 1)) 
x=l 
Since this integral exists, we know that integrating 
will give the Same answer. We demonstrate this: 
Y 
I = Jol dx 1' dy (xy2 + 1)2 
1 
= A l l d x l l d z 2 
(22 + 1)2 
first with respect to y 
x=l 1 
2 
= - log(1 + x) 
= ; log2. 
15. Changing Order of Integration 63 
Notes 
[l] The Order of integration can make a difference if Fubini’s theorem is not 
satisfied. As an example, define the function f(x, Y) = (2’ - y2))/(x2 + y2)’, 
and consider integrating it over the unit Square. Depending on the Order 
of the integrations, we will obtain different answers. Integrating 
respect to y results in 
If, instead, we integrated first with respect to x we find 
first with 
y=l 
$/=O 
ix=1 
[2] Other examples where the requirements for Fubini’s theorem are not satisfied 
are easy to find. For example, 
while 
Another example is given by 
1’ dy lw (e-xy - 2e-2xy) dx # Lm dx 1’ (e-xy - 2e-2z9) dy. 
[3] The technique of interchanging integration Order can be used to derive 
analytical formulae for int egrals. 
s x y dx dy can be evaluated in two 0 For example, the integral I = 
O<x<l 
a 5 y 5 b 
different ways, 
I = / b d y J l ” d x = J Y b - = l o g ( E ) l + b 
Y + l 
1 b x - x a 
dx 
64 I1 Concepts and Definitions 
dx = log (-) l + b 
i + a ’ 
dx dy dz 
to yield the result 
0 As a second example, the integral s 1 s can 
O < z , y < l ( 1 + x2z2)(1 + y2z2) 
2 2 0 
be used to derive the relation (See George [2 ] , page 206) 
Lw (q)2 dz = 7rlog2. 
[4] A more general Statement of Fubini’s theorem is: 
Theorem: (Fubini) Let X = Rp and Y = Rp; then the 
formula 
is valid in each of the following two cases: 
[ l ] 
[2] 
f is a measurable positive arithmetic function on X x Y ; 
f is an integrable function over X x Y . 
References 
[ l ] 
[2] 
[3] 
G. F’ubini, “Sugli Integrali Multipli,” Opere Scelte, Cremonese, Vol 2, 1958, 
pages 243-249. 
C. George, “Exercises in Integration,’’ Springer-Verlag, New York, 1984. 
N. I. Muskelishvili, Singular Integral Equations, Noordhoff, Groningen, 1953, 
pages 56-61. 
16. Convergence of Integrals 
Applicable to Single and multiple integrals. 
Yields 
Knowledge of whet her an integral converges or diverges. 
Idea 
or divergence of the original integral may be determined. 
By comparing a given integral to a different integral, the convergence 
16. Convergence of Integrals 65 
Procedure 
Most techniques that indicate convergence or divergence use some sort 
of integral inequality (see Page 205). In this section we use the following 
two theorems: 
Theorem (Comparison test for convergence): Let f(z) and 
g(z) be continuous for a < z 5 b, with 0 5 If(z)l 5 g(z). If 
Jab g(z) dz converges, then s,” f(z) da: converges and 
Theorem (Comparison test for divergence): Let f(z) and 
g(z) be continuous for a < z 5 b, with 0 5 g(z) I f(z). If 
s,” g(z) dz diverges, then s,” f(z) dz diverges. 
To use these theorems effectively, a knowledge base must be created 
of converging and diverging integrals. Some common integrals used for 
comparison include (assuming U < b): 
, which converges for p < 1 and diverges for p 2 1. 
dx 
dx 
(B) 1 w, which converges for p < 1 and diverges for p 2 1. 
(c) J1e 
O0 dx 
(D) J zp7 
(E) j- dx 
which converges for p < 1 and diverges for p 2 1. 
which converges for p > 1 and diverges for p 5 1. 
which converges for p > 1 and diverges for p 5 1. 
- 1 xp ’ 
2 x(l0gx)P’ 
Example 1 
For the given range of 
integration (i.e., for x E [O,m]) we can easily bound the trigonometric 
term: Ix sin2 X I 5 x . The integral J = s-”, & dx will now be shown to 
converge, which then implies the convergence of I . First, we write 
Consider the integral I = s-”, z,$:T dx. 
The integrand in J1 is bounded above by x , and J i x d x = 3 (this 
shows that J1 is convergent). The integrand in J2 can be bounded above 
by x - ~ and Jr x - ~ dx = 1 (this shows that J2 is convergent). Hence, J is 
convergent (it is bounded above by 4) and so is I . 
66 I1 Concepts and Definitions 
Example 2 
Consider t he integral 
I = lm (La sin(y2x3) dy dx. ) 
I = 1 2 lm ( / a 2 z 3 &) dz x3/2 * 
Changing variables in the inner integral to x = y2x3 results in 
The inner integral is bounded for all U and x since 3 dx converges. 
The integral Jlm x-3/2 dx also converges, so we conclude that I converges. 
Notes 
[l] Note that we can analytically integrate J in Example 1. We find that 
X 
J = Ja;- dx 
-00 x3 + 1 
5= 00 
1 1 - x + x 2 1 1 2 2 - 1 +-tan- -)I 
J3 x = o 
= (-$og 
( l + x ) 2 J3 
[2] Another useful theorem for determining whether an integral converges is 
Chartier's test (See Whittaker and Watson [3]) 
Theorem (Chartier): If f ( x ) decreases to Zero monotoni- 
cally as x -+ 00, and 11: 4(t) d t ( is bounded as x -+ 00, then 
s," f ( x ) 4 ( x ) dx converges. 
(A) For example, consider I = JOw 2-l sin x dx. In this case, f ( x ) = 2-l is 
monotonically decreasing and Jo5 sin t d t = 1 - COS t , which is certainly 
bounded. We conclude that I converges. 
(B) For a more interesting example, consider J = Jla; x cos(x3 -2) dx. This 
integral can be written as J=lOOL(d - sin(x3 - x ) ) dx. 
3x2 - 1 dx 
We recognize that x / (3x2 - 1) decreases monotonically, and that 
which is bounded. We conclude that J converges. 
[3] The following decomposition 
sin(a - 1)x 00 
K = L W X da: = f 1 X + ' I x d x + ; l m dx , COS x sin ax 
combined with the first example in the last note Shows that K converges. 
Divergent integrals can sometimes be regularized to obtain a finite value, 
See Wong and McClure [2]. 
[4] 
17. Exterior Calculus 67 
Re ferences 
[ 11 
[2] 
[3] 
W. Kaplan, Advanced Calculus, Addison-Wesley Publishing Co., Reading, 
MA, 1952. 
R. Wong and J. P. McClure,“Generalized Mellin Convolutions and their 
Asymptotic Expansions,” Can. J. Math, 36, 1984, pages 924-960. 
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cam- 
bridge University Press, New York, 1962. 
17. Exterior Calculus 
Applicable to 
Idea 
The integral of a differential form is a higher-dimensional generaliza- 
tion of such ideas as the work of a force along a path, or the flux of a fluid 
across a surface. 
Integration of differential forms. 
Procedure 
A real-valued function of x, also known as a scalar field, is called a 
zero-form in exterior calculus. A one-form (in, say, three space) is an 
expression of the form FI dx + F2 d y + F3dz, where Fl , F2, and F3 are 
functions of x , Y , and z. 
A two-form in “Standard Order” is an expression of the form 
The inverted v’s are called carets or wedges. The wedges distinguish be- 
tween a two-form, such as 4y3dz A dx, and objects such as 4y3dzdx, 
that occur in double integrals where the Order of the differentials is not 
important. 
In n-dimensional space, p-forms exist for p = 0, 1, . . . , n. The addition 
of two p-forms, or the multiplication of a p-form by a scalar, is another 
p-form. 
The wedge product has the following properties: 
0 The wedge product is associative: w A (8 A () = (w A 8) A (. 
0 The wedge product is anti-commutative: w A 8 = (-1)“8 A u , for w a 
0 The wedge product of repeated differentials vanishes: dx A dx = 0 
0 The wedge product of a p-form and a q-form is a ( p + q)-form. 
k-form and 8 an Z-form. 
When dealing with n-dimensional space, it is useful to deal with forms 
in a more formal way. If we choose a linear coordinate system { X I , 2 2 , . . . , xn} 
on Rn, then each xi is a 1-form. If the { x i } are linearly independent, then 
every 1-form w can be written in the form w = alxl + . . . + a,x„ where 
68 I1 Concepts and Definitions 
the { u i } are real functions. The value of w on the vector C is equal to 
w(C) = a1x1(C)+. . .+anzn(C), where x l ( C ) , . . . , xn(C), are the components 
of C in the Chosen coordinate System. 
A 2-form w2 is a function on pairs of vectors which is bilinear and 
skew-symmetric: 
From this it is easy to derive that wa(C, ( ) = 0 for every 2-form. Analo- 
gously, an exterior form of degree k , or a k-form, is a function of k vectors 
which is k-linear and antisymmetric. 
A differentiable k-form wkIx, at a Point x of a manifold M , is an 
exterior k-form on the tangent space T M x to M at the Point x. That is, 
it is a k-linear skew-symmetric function of k vectors (61 , . . . , Ck} tangent 
to M at x. Every differentiable k-form on the space R" with a given 
coordinate System ( ~ 1 , . . . , xn} can we written uniquely in the form 
where the {u i l , . . . , ik (x) } are smooth functions on R". 
A chain of dimension n on a manifold M consists of a finite collection 
of n-dimensional oriented cells 0 1 , . . . , 0,. in M and integers m l , . . . , m,., 
called multiplicities (the multiplicities can be positive, negative, or Zero). 
A chain is denoted by ck = m l a i + . . . + m,.a,.. 
Let ( 5 1 , . . . , xl} be an oriented coordinate System on Rk. Then every 
k-form on Rk has the form wk = +(x) d q A s e .Adxk, where +(x) is a smooth 
function. Let D be a bounded convex polyhedron in Rk. We define the 
integral of the form wk on D to be 
wk = +(x) d x l . . . dxk, 
where the integral on the right is understood to be the usual limit of 
Riemann Sums. To integrate a k-form over an n-dimensional manifold, 
the role of the usual path of integration is replaced by a k-dimensional cell 
o of M represented by a triple 0 = (D, f , Or) where 
(A) D c Rk is a convex polyhedron; 
(B) f : D + M is a differentiable map; 
17. Exterior Calculus 69 
(C) Or represents an orientation of D on Rk. 
Then the integral of the k-form w over the k-dimensional cell U is the 
integral of the corresponding form over the polyhedron D: Ju w = JD f * w . 
Here the form f * w is defined by 
where the {Ci} are tangent vectors and f* is the differential of the map f . 
The integral of the form w k over the chain ck is the sum of the integrals on 
the cells, counting multiplicities: Jck wk = Ci mi Jui wk. 
Some classes of integrals can be immediately evaluated. Using the two 
definitions: 
0 A differential form w on a manifold M is closed if its exterior derivative 
is Zero: dw = 0, 
0 A cycle on a manifold M is a chain whose boundary is equal to Zero, 
we have the two theorems: 
The integral of a closed form w k over the boundary of any (k + 1)- 
dimensional chain c k + l is equal to Zero: That is, wk = 0 if 
dwk = 0; 
The integral of a differential over any cycle is equal to Zero: That is, 
Jck+l d u k = 0 if dck+l = 0. 
Example 
The two-form T = ( x d x + y d z j A (Y d x - y 2 d y ) can be expanded into 
T = x y d x A d x i- y 2 d z A d x - x y 2 d x A d y - y3 d z A d y . 
The first term vanishes because it contains a repeated differential. Writing 
the remaining terms in Standard Order results in 
T = y 3 d y A d z + y 2 d z A d x - ~ y 2 d x A d y . 
Notes 
[l] 
[2] 
As a matter of convention, the function identically equal to Zero is called a 
p-form for every p. 
Stokes' formula can be stated as Jacw = Jcdw, where c is the (k + 1)- 
dimensional chain on a manifold M and w is any k-form on M . 
70 I1 Concepts and Definitions 
References 
[l] L. Arnold, Mathematical Methods of CZassical Mechanics, translated by K. 
Vogtmann and A. Weinstein, Second edition, Springer-Verlag, New York, 
1989, Chapter 7. 
B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern Geometry 
Methods und Applications, Springer-Verlag, New York, 1991. 
D. G. B. Edelen, Applied Exterior Calculus, John Wiley & Sons, New York, 
1985. 
P. J. Olver, Applications of Lie Groups to Diflerential Equations, Graduate 
Texts in Mathematics #107, Springer-Verlag, New York, 1986. 
H. Whitney, Geometric Integration Theory, Princeton University Press, Prince- 
ton, NJ , 1957. 
[2] 
[3] 
[4] 
[5] 
18. Feynman Diagrams 
Applicable to 
ordinary or pat h integrals. 
Feynman diagrams are used to denote a collection of 
Idea 
of integrals. The diagrams are used to keep track of the terms. 
Some integrals equations have a “natural” expansion scheme in terms 
Procedure 
If a given differential equation is only a “small” Perturbation from 
a linear differential equation (with a known Green’s function), then we 
may obtain an equivalent integral equation. This integral equation may be 
expanded methodically into a series of integrals. Diagrams are often used 
to keep track of the terms. 
Example 
Consider t he nonlinear ordinary differential equation 
z (0 ) = 0. 
in which the nonlinear term (i.e., the g ( t ) function) is “small.” 
equation may be directly integrated to obtain 
This 
(18.1) 
18. Feynman Diagrams 71 
- H ( t - 7 ) 
Figure 18.1 Rules for creating diagrams and rules for interpreting diagrams. 
If the value of x ( t ) from the left hand side of (18.1) is used in the right 
hand side, t hen 
A “natural” Perturbation expansion would be to keep the first two 
terms in the right hand side of (18.2), and assume that the last two terms 
are “small”. If Iz( t ) l < 1 then this may well be the case since the last two 
terms involve 1zl2 while the first two terms involve constants. 
A functional iteration technique can be used to derive (18.2) and its 
higher Order extensions from diagrams. We need two sets of rules: One 
set of rules describes how the diagrams may be computed; the other set 
of rules describes how the diagrams are to be turned into mathematical 
expressions. If we use the rules in Figure 18.1, (where H ( ) denotes the 
Heaviside function), then the first two steps in the diagrammatic Solution 
to z ( t ) (from (18.1)) are given by the diagrams in Figure 18.2. 
Note that the third and fourth diagrams in Figure 18.2 represent the 
Same mathematical expression since they are topologicallyequivalent . The 
purpose of the Heaviside function is to restrict the range of integration. By 
careful inspection, the mathematical expressions associated with the last 
set of diagrams will be seen to be identical to (18.2). See Zwillinger [5] for 
more details. 
72 I1 Concepts and Definitions 
Figure 18.2 Two steps in the diagrammatic expansion of (18.1). 
Notes 
[l] Often an “algebra of diagrams” is created, so that diagrams can be added, 
subtracted and multiplied without recourse to the mathematical expression 
that each diagram represents. This would require amplification of the rules 
that were used in the example. 
This technique is particularly important in Problems in which there is no 
“small” Parameter. In these cases, the formally correct diagrammatic expan- 
sion may be algebraically approximated by exactly summing certain classes 
of diagrams. See Mattuck [4] for details. 
[2] 
References 
F. Battaglia and T. F. George, “A Rule for the Total Number of Topologi- 
cally Distinct Feynman Diagrams,” J. Math. Physics, 25, No. 12, December 
1984, pages 3489-3491. 
T. Kaneko, S. Kawabata, and Y. Shimizu, “Automatic Generation of Feyn- 
man Graphs and Amplitudes in QED,” Comput. Physics Comm., 43, 1987, 
pages 279-295. 
J. Küblbeck, M. Böhm, and A. Denner, “Feyn Arts - Computer-Algebraic 
Generation of Feynman Graphs and Amplitudes,” Comput. Physics Comm., 
60, 1990, pages 165-180. 
R. D. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem, 
Academic Press, New York, 1976. 
19. Finite Part of Integrals 73 
[5] D. Zwillinger, Hundbook of Diflerentiul Equutions, Academic Press, New 
York, Second Edition, 1992. 
19. Finite Part of Integrals 
Applicable to Divergent integrals. 
Yields 
Idea 
diverging piece of an integral, a finite term is left. 
Procedure 
A convergent expression. 
Many divergent integrals diverge in a common way. By removing the 
Given the integral 
I = [(x - a)'-"f(x) dx, (19.1) 
we assume that f ( a ) # 0 and -1 < X 5 0. Clearly, this integral diverges 
for all integral value of n. 
However, when f E C"[s,r] and X # 0, we define the finite Part of I 
to be the value of 
(19.2) 
b ;:+ [J1 (5 - a)x -"f (5 ) dz - g ( t ) ( t - @"+l 
where g ( t ) is any function in Cn[s7r ] such that the above limit exists. 
Denote the finite Part of the integral I by f,"(x - ~ ) ~ - " f ( x ) dx. The limit 
in (19.2) can be explicitly evaluated to find (for X # 0): 
For X = 0 we find: 
74 I1 Concepts and Definitions 
Some Special cases are easy to express in terms of a Cauchy principal 
value type limit. For example, if W ( X ) is a non-negative weight function 
integrable on the interval ( U , b ) , c is in the interval ( U , b) , and f(x) is dif- 
ferentiable in a neighborhood of c with its derivative satisfying a Lipschitz 
condition, then (See Paget [8]) 
Example 
Consider the finite-part integral J = ixy dy. This can be written 
as 
For finite-Part integrals, we have the usual relations 
#x 7 = -: and ix = logz. 
O Y 
(19.6) 
Combining these results, and assuming that f(y) possesses a continuous 
second derivative on the interval [ O , X ] , we have 
Notes 
[l] 
[2] 
This is also called Hadarnard's f inite part of a n integral. 
The finite parts of integrals have different properties from usual integrals. 
Some of these properties are (from Davis and Rabinowitz [2]): 
f is a consistent extension of the concept of regular integrals. 
f is additive with respect to the Union of integration intervals and is 
invariant with respect to translation. 
f is a continuous linear functional of the integrand. 
f ö z - " d x = 0 if CU > 1. 
fö z-l dz = 0 (this is consistent with (19.6)). 
For a > 1, we have ft x-" d x = 1/(1 - CU) < 0. Hence, if f(z) > 0 , it 
may be that . f f (z)dx < 0. 
[3] Kutt [7] has derivid quadrature rules for the evaluation of finite-part inte- 
grals of the form f(z) dz, when s E (a ,b ) and lc = 1 , 2 , . . .. 
20. Fractional Integration 75 
[4] It is possible to write some finite-Part integrals as derivatives of principal- 
value integrals. For example: 
b 
f dx = - w:yc’ dx. 
This is the basis for the numerical technique in Paget [8]. 
Davis and Rabinowitz (21 give the example [5] 
” d5 - 2J5 [log ( l O & - 30) - 11 - 5. 
[6] Other classes of finite-Part integrals also exist. 
theories developed for each of the integrals 
For example, there are 
where m is a positive integer. 
Finite part integrals arise naturally in fluid mechanics, solid mechanics, and 
electromagnetic t heory. See, for example, Ioakimidis [4]. 
[7] 
References 
B. Bialecki, “A Sinc Quadrature Rule for Hadamard Finite-Part Integrals,” 
Numer. Math., 57, 1990, pages 263-269. 
P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second 
Edition, Academic Press, Orlando, Florida, 1984, pages 11-15 and 188-190. 
N. I. Ioakimidis, “On the Gaussian Quadrature Rule for Finite-Part Integrals 
with a First-Order Singularity,” Comm. Appl. Numer. Meths., 2, 1986, pages 
N. I. Ioakimidis, “Application of Finite-Part Integrals to the Singular Inte- 
gral Equations of Crack Problems in Plane and Three-Dimensional Elastic- 
ity,” Acta. Mech., 45, 1982, pages 31-47. 
N. Ioakimidis, “On Kutt’s Gaussian Quadrature Rule for Finite-Part Inte- 
grals,,, Appl. Num. Math., 5, No. 3, 1989, pages 209-213. 
N. I. Ioakimidis and M. S. Pitta, “Remarks on the Gaussian Quadrature 
Rule for Finite-Part Integrals with a Second-Order Singularity,” Computer 
Methods in Appl. Mechanics und Eng., 69, 1988, pages 325-343. 
H. R. Kutt, “The Numerical Evaluation of Principal Value Integrals by 
Finite-Part Integration,” Numer. Math., 24, 1975, pages 205-210. 
D. F. Paget, “The Numerical Evaluation of Hadamard Finite-Part Inte- 
grals,,, Numer. Math., 36, 1981, pages 447-453. 
G. Tsamasphyros, and G. Dimou, “Gauss Quadrature Rules for Finite Part 
Integrals,” Internat. J. Numer. Methods Engrg., 30, No. 1, 1990, pages 13- 
26. 
123-1 32. 
20. Fractional Integration 
Applicable to Fkactional integrals. 
76 I1 Concepts and Definitions 
Procedure 
The Riemann-Liouville fractional derivative of Order v is defined by 
(This is sometimes represented by &,) The fractional derivative has the 
follow ing propert ies: 
The Operation of Order Zero leaves a function unchanged: c D z f ( z ) = 
The law of exponents for integration of arbitrary Order holds: cDZ 
The fractional Operator is linear: cD:[af (z ) + b g ( x ) ] = a,DZf(x) + 
b , D : g ( x ) (assuming, for v < 0, that f and g are analytic). 
The Operation cDZf ( z ) yields f(")(z), the v-th derivative of f , when 
v is a positive integer. If v is a negative integer, say v = -n, then 
cD,nf(x) is the Same as the ordinary n-fold integration of f(z). (The 
integration constants are Chosen so that cD;nf(x) vanishes, along 
with its first n - 1 derivatives, at x = c.) 
If f (z) is an analytic function of the complex variable z , the function 
J3Zf(z) is an analytic function of v and x. 
f (x>. 
f(4 = c q ' " f ( z ) . 
Many common functions can be written as fractional integrals of other 
functions. For example, we find 
It is also possible to represent some ordinary int egrals as fract ional 
integrals in non-obvious ways. For example, from Oldham and Spanier [3] 
we have (page 182) 
or, when x = 1 
20. Fkactional Integration 77 
For example, we find 
L’sin (JS) 
= J.l ( d a 1 (&))I,=, 2 
n- 
= 4 1 (1) 
2 
= .69122984.. . . 
Notes 
[l] 
[2] 
A semi-integral is a half integral, that is w = - 9 in (20.1). Table 20 contains 
a short table of semi-integrals. 
Another definition of a fractional derivative is Weyl’s integral 
( t - x)’-’f ( t ) dt 
for w > 0. To compute fractional integrals, let m be the smallest positive 
integer such that < m and define r = m - w. Then 
Note that we also have 
= - d” (1 l x ( x - t)‘-’ f ( t ) d t ) 
dx” r ( r ) 
when m is the smallest positive integer such that w < m and T = m - w. 
Osler [4] has established the fractionalintegral generalization of Leibniz’s 
rule: 
[3] 
where 7 is arbitrary. 
Refer ences 
A. Erdelyi, “Axially Symrnetric Potentials and F’ractional Integration,” J. 
SOC. Indust. Appl . Math., 13, No. 1, March 1965, pages 216-228. 
F. G. Lether, D. M. Cline and 0. Evans, “An Error Analysis for the Cal- 
culation of Semiintegrals and Semiderivatives by the RL Algorithm,” Appl. 
Math. und Comp., 17, 1985, pages 45-67. 
K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, 
New York, 1974. 
T. J. Osler, “Leibniz Rule for F’ractional Derivatives Generalized and an 
Application to Infinite Series,” SIAM J . A p p l . Math., 18, No. 3, May 1970, 
pages 658-674. 
B. Ross, Fractional Calculus and Its Applications, Proceedings of the Inter- 
national Conference at the University of New Haven, June 1974, Springer- 
Verlag, New York, Lecture Notes in Mathematics #457, 1975. 
78 I1 Concepts and Definitions 
Table 20. A short table of semi-integrals. 
0 
C, a constant 
x-112 
X 
xn, n = 0,1,2, 
xp, p > -1 
dTTT 
d E 
1 
1 
l + x 
ex 
ex erf (J.) 
sin (J.) 
COS (J.) 
sinh (J.) 
cosh (J.) 
sin (J.) 
J. 
J. 
COS (&) 
log x 
log x 
J. 
0 
2 c f 7r 
(n!)2 (4xy+lI2 
(2n + l)!fi 
2 - arctan (J.) 
fi 
2 sinh-' (&) 
l/m 
ex erf (J.) 
ex - 1 
f i J 1 (J.) 
f i H - i (J.) 
-11 (J.) 
f i L - 1 (J.) 
f i J 0 (J.) 
21. Liouville Theory 
21. Liouville Theory 
Applicable to Indefinite one-dimensional 
Yields 
79 
integrals. 
Knowledge of whether the integral can be integrated in closed form in 
terms of “elementary functions.” 
Idea 
The elementary functions are defined to be 
[i] rational functions, 
[2] algebraic functions (explicit and implicit), 
[3] exponential and logarithmic functions, 
[4] functions generated by a finite combination of the preceding classes. 
The derivative of an elementary function is an elementary function. 
The question that Liouville addressed is: “When is the integral of an 
elementary function an elementary function?” 
Procedure 
the following definitions: 
The theory underlying the evaluation of integrals is complex. We need 
0 Let K be a field of functions. The function 0 is an elementary generator 
- 0 is algebraic over K , i.e., 0 satisfies a polynomial equation with 
- 0 is an exponential over K , i.e., there is a < in K such that 0‘ = C’0, 
- 8 is a logarithm over K , i.e., there is a in K such that 6’ = <’/<, 
0 Let K be a field of functions. An over-field K ( 01, . . . , 0,) of K is called 
a field of elementary functions over K if every 0i is an elementary 
generator over K . A function is elementary over K if it belongs to a 
field of element ary func t ions over K . 
Note that, with K = C (the complex numbers), trigonometric func- 
tions (and their inverses) as well as rational functions are elementary. For 
example cos z = (eiz + e-2’ ). That is, the cosine function is made up of 
complex multiplications, complex exponentiations, and rational Operations. 
Theorem: (Liouville’s principle) Let f be a function from a 
function field K. If f has an integral elementary over K, it 
has an integral of the following form: 
over K if 
coefficients in K ; 
which is an algebraic way of saying that 0 = exp<; 
which is an algebraic way of saying that 0 = log<. 
The fundamental theorem in this area is 
J f = 210 + C c i i o g v i , 
i= 1 
80 I1 Concepts and Definitions 
h 
where W O belongs to K , the {wi} belong to K , an extension 
of K by a finite nuFber of constants algebraic over K , and 
the { c i } belong to K and are constant. 
For example: 
If an integral of an algebraic function y is elementary, then the integral 
must be of the form 
where the {Rk} are all rational functions and the number of terms in 
the sum is finite but undetermined. 
If y is given by the Solution of an Nth-degree polynomial equation, 
and the integral of y is purely algebraic, then the integral must be of 
the form 
N-1 
1 9 = Rk(z)yk 
where the {Rk} are all rational functions. 
If the integral I = J f e g da: is elementary, and f and g are elementary, 
then I is of the form I = Res, where R is a rational function of x, f, 
and 9. 
Risch proved an extension to the above theorem that incorporates 
logarithms and exponentials: 
Theorem: (Risch) Let K = C(z, 01,192,. . . ,On) be a field of 
functions, where C is a field of constants and each Bi is a loga- 
rithm or an exponential of an element of C(z, 81,82,. . . , &-I) 
and is transcendental over C(z, 81, 82, . . . , &-I). Moreover, 
the field of constants of K must be C. There is an algorithm 
which, given an element f of K , either gives an elementary 
function over K that is the integral of f or proves that f has 
no elementary integral over K . 
Davenport proved an extension to the above theorem that incorporates 
some algebraic functions: 
Theorem: (Davenport) Let K = C(z, y, 81,&,. . , , 0,) be a 
field of functions, where C is a field of constants, y is algebraic 
over C(z), and each 8i is a logarithm or an exponential of an 
element of C(z, y, 81,192,. . . , &--I), and is transcendental over 
C(z, y, 81,82, . . . , 8i-1). Moreover, the field of constants of 
K must be C. There is an algorithm that, given an element 
f of K , either gives an elementary function over K that is 
the integral of f or proves that f has no elementary integral 
over K . 
21. Liouville Theory 81 
There is, as yet, no theorem that will allow algebraic extensions that 
depend on logarithms or exponential functions. Note that the theorems 
above are for integrating elementary functions when the integrals also have 
to be elementary. There are two ways of expanding the above results: 
0 Allow a larger or different class of functions to appear in the integrand 
(in this case the class of functions allowed in the evaluated integral 
may be the Same or different). 
0 Allow a larger or different class of functions to appear in the eval- 
uated integral, such as those created from elementary functions and 
logarithmic integrals, error functions, or dilogarithms. 
See Baddoura [ l ] , Cherry [3] and [4], and Davenport, Siret, and Tournier [6] 
for details. 
Example 1 
Using the above Statements, we can resolve the question “1s the integral 
I = s e x 2 / x d x elementary?” If it is, then by Statement number 3 (above), 
the integral must be of the form I = Rex2 where R is a rational function 
of x , x-l, and x 2 . That is, R is a rational function of x . Equating the two 
expressions for I , and differentiating, results in 
ex2 - = R’ex2 + 2xRex2 or x(R’ + 2xR) = 1 . ( 2 1 . 1 ) 
X 
Since R is a rational function of x, it can be written in the form R = 
N ( x ) / D ( x ) , where N ( z ) is a polynornial of degree n and D ( x ) is a polyno- 
mial of degree d. Evaluating the differential equation for R in (21.1), using 
N and D , results in 
xDN’ - xD’N + 2x2ND = D2. ( 2 1 . 2 ) 
Each term in this equation is a polynomial, the degrees of the terms are 
d + n, d + n, d + n + 2, and 2d. The first two terms in ( 2 1 . 2 ) cannot 
equal each other unless N = D (which implies that R is a constant, which 
does not work). Clearly, the third term cannot have Same degree as the 
first term. The fourth term cannot balance both the first and third terms, 
so no values of n and d will work. We conclude that I does not have an 
element ary integral. 
82 I1 Concepts and Definitions 
Example 2 
As another application of Statement number 3, we investigate the 
integral I = s e-" dz. If this integral is elementary, then the integral must 
be of the form I = Re-x2 where R is a rational function of x. Equating 
the two expressions for I and differentiating results in 
e - X 2 - - Rle-X2 + Re-x2 (-2z). 
This can be written as 1 = R' - 2 R x . If we write R(z) = P(z ) /Q(z ) , 
where P ( z ) and Q(z ) are relatively prime polynomials, then we obtain the 
relat ion 
Q(Q - P' + 22P) = -PQ'. (21.3) 
Q(z ) must have at least one root, ,ß. Mence, Q(z ) can be written inthe 
form Q(z ) = ( ~ - , ß ) ~ T ( z ) , where T is a positive integer and T(P) # 0. Since 
Q and P are relatively prime, P(@) # 0. Now we have a contradiction: ,ß 
is a root of the left-hand side of (21.3), with a multiplicity of at least r , but 
,ß is a root of the right-hand side of (21.3), with a multiplicity of at most 
T - 1. Hence, the assumption that Se-"' dx has an elementary integral 
must be incorrect. 
Example 3 
Consider the integral I = Sze-"' d z . Assuming the integral has the 
form I = Re-x2 results in the following differential equation: R'-2zR = x. 
This has the Solution R = -$, so s X e b x 2 dz is integrable. 
Notes 
The symbolic Computer language MAPLE will optionally, in the Course of 
trying to evaluate an indefinite integral, print out information similar to the 
steps in the above examples. This makes it easy to monitor the computation 
and verify the result. 
The integration procedures described in this section are actually used in 
Computer languages that can perform indefinite integration. See the section 
beginning on page 117. 
Even if an indefinite integral is not elementary, it sometimes possible to 
evaluate a definite integral with the Same integrand. For example, even 
though dx is not elementary (See Example 2), the definite integral 
S-, -z2 dx is easily shown to be equal to fi (See for example, page 115). 
Sometimes, in practice, it seems as if most integrals of interest are not 
elementary. It is certainly true that many simple-looking integrals are 
not elementary, such as J&, Je dx, Jsinx2 dx, J d G - d x , and 
s d m d x . 
00 
22. Mean Value Theorems 
References 
83 
J. Baddoura, “Integration in Finite terms and Simplification with Diloga- 
rithms: A Progress Report,” in E. Kaltoflen and S. M. Watt (eds.), Comput- 
ers und Mathematics, Springer-Verlag, New York, 1990, pages 166-181. 
M. Bronstein, “Symbolic Integration: Towards Practical Algorithms,” in 
Computer Algebra und Diflerential Equations, Academic Press, New York, 
1990, pages 59-85. 
G. W. Cherry, “An Analysis of the Rational Exponential Integral,” SIAM 
J. Comput, 18, No. 5, October 1989, pages 893-905. 
G. W. Cherry, “Integration in Finite Terms with Special Functions: The 
Error Function,” J. Symbolic Comp., 1, 1985, pages 283-302. 
J. H. Davenport, “The Risch Differential Equation Problem,” SIAM J. Com- 
put, 15, No. 4, November 1986, pages 903-918. 
J. H. Davenport, Y. Siret, and E. Tournier, Computer Algebra: Systems und 
Algorithms for Algebraic Computation, Academic Press, New York, 1988, 
pages 165-186. 
K. 0. Geddes and L. Y. Stefanus, “On the Risch-Norman Integration Method 
and Its Implementation in Maple,” Proceedings of ISSAC ’89, ACM, New 
York, 1989, pages 218-227. 
P. H. Knowles, “Integration of Liouvillian Functions with Special Func- 
tions,” SYMSAC ’86, ACM, New York. 
J. Moses, “Symbolic Integration, the Stormy Decade,” Comm. ACM, 14, 
1971, pages 548-560. 
R. D. Richtmyer, “Integration in Finite Terms: A Method for Determining 
Regular Fields for the Risch Algorithm,” Lett. Math. Phys. 10, No. 2-3, 
1985, pages 135-141. 
R. H. Risch, “The Problem of Integration in Finite Terms,” Trans. Amer. 
Math. SOC., 139, 1969, pages 167-183. 
R. H. Risch, “The Solution of the Problem of Integration in Finite Terms,” 
Bull. Amer. Math. SOC., 76, 1970, pages 605-608. 
J. F. Ritt, “On the Integrals of Elementary Functions,” Trans. Amer. Math. 
SOC., 25, 1923, pages 211-222. 
J. F. Ritt, Integration in Finite Terms: Liouville’s Theory of Elementary 
Methods, Columbia University Press, New York, 1948. 
W. Squire, Integration for Engineers und Scientists, American Elsevier Pub- 
lishing Company, New York, 1970, pages 41-49. 
22. Mean Value Theorems 
Applicable to Single and multiple integrals. 
Yields 
Information about a function integrated over a region. 
84 I1 Concepts and Definitions 
Idea 
that function times the integral of a simplier function. 
The integral of a function can sometimes be written as some value of 
Procedure 
value of a quantity; we indicate only a few of them. 
There are several theorems that yield information about the mean 
First Mean Value Formula 
Assuming that a 5 b, we have 
If f(z) is continuous then we have 
where a 5 2 5 b. 
(22.1) 
This theorem is sometimes written as (assuming, again, that f(z) is con- 
t inuous) 
Second Mean Value Formula 
If f is a decreasing positive function on [a,b], and g an 
integrable function on this interval, then 
Mean Value Theorem for Double Integrals 
If f ( z ,y ) is continuous on a compact region R, with area A, 
there exists a Point ( c , ~ ) in the interior or R such that 
(22.2) 
22. Mean Value Theorems 85 
Example 
Consider the integral I = J: x sina: dz. The exact evaluation is I = 
sinx - X C O S X ~ ; = T. From (22.11, we can write I = ~ ~ s i n ~ , for at least 
one value of Z in the range [0, T I . In this case, we find that 2 N 1.1141 is 
one such value. 
Notes 
[l] There are two theorems that are also called the second mean value theorems 
(See Gradshteyn and Ryzhik [2], page 211): 
(A) If f(z) is monotonic and non-negative in the interval ( U , b) (with U < b), 
and if g(z) is integrable over that interval, then there exists at least 
one Point [ in the interval such that 
(B) If, in addition to the requirements in the last Statement, f(z) is non- 
decreasing, then there exists at least one Point C in the interval such 
t hat 
[ f (zM4 dz = f ( b ) Jllg(4 dz. 
[2] There is also a mean value theorem from complex analysis (See Iyanaga and 
Kawada [3], page 624): 
Let U be a harmonic function (i.e., V2u = 0), let D be the 
domain of definition of U, and let S be the boundary of D. 
Then the mean value of U on the surface or the interior of 
any ball in D is equal to the value of U at the Center of the 
ball. That is 
where rn and cn are the volume and surface area of a unit 
ball in Rn, B ( P , r ) is the Open ball with Center at P and 
radius r , S(P, r ) is the spherical surface with Center at P and 
radius r , d r is the volume element, and da is an element of 
surface area. 
References 
[l] 
[2] 
[3] 
W. Kaplan, Advanced Calculus, Addison-Wesley Publishing Co., Reading, 
MA, 1952. 
I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, und Products, 
Academic Press, New York, 1980. 
S. Iyanaga and Y. Kawada, Encyclopedic Dictionary of Mathematics, MIT 
Press, Cambridge, MA, 1980. 
86 I1 Concepts and Definitions 
23. Path Integrals 
Idea 
Path integrals are integrals of functionals. Feynman path integrals are 
specific pat h integrals for t he propagator of t he Schrödinger equation; t hey 
have the form G = Jeas/’, where S is the action. 
Procedure 1 
We illustrate the general procedure by considering a specific case. 
Consider a collection of particles that perform Brownian motion in 
one dimension and do not interact with one another. Define c(z , t )dz to 
be the number (or concentration) of particles in an interval of size dz about 
the Point z at time t. It is well known that the concentration satisfies the 
partial differential equation 
(23.1) 
where D is the diffusion coefficient. 
If one particle starts out at 20 at time t o , then the concentration has 
the initial condition c(z0, t o ) = S ( t - t o ) . (The function S( ) denotes Dirac’s 
delta function.) The Solution of (23.1) with this initial condition is (see 
Zw illinger [ 1 I]) : 
(23.2) 
This gives the likelihood of finding the particle at the Point x at 
any time t 2 t o . The likelihood of finding the particle near z, with an 
uncertainty of dz, is just c(z, t ) dx. (In mathematics books, c(z, t ) would 
be called the Green’s function of (23.1); in physics books it would be called 
the propagator of a Brownian particle.) 
Now divide the interval ( t o , t ) into N + 1 equal intervals of length E: 
tl < t 2 < . . . < t~ (so that ti+1 - ti = E ) . We would like to know the 
probability that the particle that started at (z0,to) is nearthe Point z1 
(with uncertainty dz1) at time t l , near the Point z2 (with uncertainty dz2) 
at time t 2 , . . . , near the Point z~ (with uncertainty d z ~ ) at time t ~ , and 
near the Point z := z ~ + 1 (with uncertainty dz) at time t := t ~ + 1 . See 
Figure 23. Because the Brownian particle is memoryless, this probability 
is equal to the product of (23.2) taken over successive subintervals: 
(23.3) 
23. Path Integrals 87 
, 
1 
a 
t 
n 
0 
C 
1 
0 
I 1 
Figure 23. A graphical illustration of the locational constraints on the particle. 
If the limit of infinite discretization is taken (i.e., E + 0 and N + 00, 
while ( N + 1 ) ~ = t - t o ) , then (23.3) can be interpreted as the probability 
that the particle follows a particular path z ( r ) from (xo, t o ) to (z, t ) , where 
x ( ~ j ) = zj. In this limit, the exponential in (23.3) can be written in the 
compact form: 
exp{ -L[ 4 0 t o (->z d r d r } . 
Using (23.4) in (23.3) we obtain the path integral 
Prob = / x ’ t e-( -& Lot (E) d r } d[x( . r ) ] . 
20 , t o 
The value of this path integral is, however, given in (23.2): 
(23.4) 
(23.5) 
(23.6) 
Note that the particular path taken, x ( ~ ) , does not enter into the path 
integral in (23.5). This is because the integral is considering all possible 
paths from the initial point to the final point. 
It is wortfi emphasizing that (23.5) is only a notational way to indicate 
the limit of a discrete number of integrations. The terms in (23.5) can be 
individually interpreted as follows: 
(A) The Symbol d [ z ( ~ ) ] denotes nj”=, d x j and restricts the x ( r ) appearing 
in the integral to be constrained by z(t0) = xo; xj < z(t j) < z j + d x j 
(for j = 1 , 2 , . . . , N ) ; and z ( t ~ + 1 ) = x ( t ) = x. These limits for the 
x ( ~ ) function are indicated by the limits on the first integral sign. 
(B) Each d x j is to be integrated from -00 to 00, and an implicit normal- 
ization factor is present 
where E = ( t j + 1 - t j ) / N . 
88 I1 Concepts and Definitions 
(C) The exponent in (23.5) is to be interpreted as 
(D) After all the integrations have been performed, the limit N -$ 00 
(which corresponds to E + 0) must be taken. 
Procedure 2 
Consider a classical System that Starts in state 2 , at time t a and ends 
in state X b at time t b . The evolution of the System can be described by a 
Set of variables that are functions of time; these variables describe a “path” 
from state x , to state xb. The classical System will almost certainly have 
a Lagrangian 1; that depends on the path of the System L = L[path]. The 
action S is defined as the time integral of the Lagrangian 
S[path] = 1; L[path] dt. (23.7) 
In quantum mechanics, the wave function $ evolves according to $(b) = 
J P ( b I u)$(u) dx , where P(b I U) is the propagator from X a to xb. The 
propagator can be obtained by solving the Schrödinger equation ihPt = 
H P , where H is the Hamiltonian and 7i denotes Planck’s constant divided 
by 27r. The propagator can also be obtained by the Feynman path integral 
P(b I U) = / x b ’ t b exp{ iS[path]} d[path]. 
Z a ,ta 
(23.8) 
This second method of obtaining the propagator is more useful in some 
circumstances, such as in dissipative Systems that have a Lagrangian but 
not a Hamiltonian. See Wiege1 [9] for more details. The path differen- 
tial measure takes the form d[path] ii‘ AN n;’’ d x j , where AN is an 
appropriate normalization constant . 
Example 1 
We indicate two ways in which the integral 
may be evaluated. This integral is the one in (23.5), with xo = x = t o = 0, 
but here we presume that we do not know its value. The integral in (23.9) 
can be written, by the four rules given above, as 
N+1 
I = lim (-) J” -00 d s o J m -00 d z l - - a [ : d x N 
N-WCI &F& 
23. Path Integrals 89 
where E is related to N by E = t / ( N + 1). 
Example l .A 
written as (note that zo = 0 and Z N + ~ = z = 0) 
The quadratic terms appearing in the exponent in (23.10) can be 
N N 
j = O k,Z=l 
where AdN) is the tri-diagonal matrix 
‘ 2 -1 0 
-1 2 -1 
-1 2 -1 
-1 2 -1 
, o -1 2 
(23.11) 
It is straightforward to show that de t IdN) = N + 1. is 
Hermitian, its eigenvalues are real. Denoting its eigenvalues by {Aj} , the 
sum in (23.11) may be written as Ajyj, where the {gj} are linearly 
related to the {zj}. Again, since M ( N ) is Hermitian, the Jacobian of this 
transformation is unity. Hence, (23.10) may be written as 
Since 
1 1 ) = lim (-) = - 1 
det M ( N ) N+OO &iz d r n . = lim N-w 
(23.12) 
This is precisely the value in (23.6), when z = 20 = t o = 0. 
90 I1 Concepts and Definitions 
Example 1.B 
formula 
The integrations in (23.10) can also be performed one at a time. The 
can be used to evaluate the integrals sequentially. First, choose XL = XO, 
x = X I , and ZU = 2 2 . Then the x1-integral appearing in (23.10) can be 
evaluated to obtain 
4De 
This expression has the same form as a term in (23.10), but with 
E replaced by 2e throughout. Now the integration with respect to 2 3 is 
carried out, using X L = x2 and ZU = x4. Then the integrations with 
respect to 2 5 , 27, . . . , are carried out. The variables left at this Point are 
Now the integrals with respect to 2 2 , x6, x10, . . . , are carried out, and 
again the same form is obtained, but now 2~ has been replaced by 4 ~ . This 
Operation is recursively applied. (Without loss of generality, we can assume 
{xO,x2,X4,x6, ...}. 
that N + 1 is a power 
I = 
Notes 
[l] The main task of 
of two.) The end result is 
1 - 1 lim - 
N+CO 2/4nD(N + 1 ) ~ e‘ 
evaluating a path integral is to evaluate the multiple .~ 
Riemann integrals and then take the N + 00 limit. This is only possible in 
a few cases, such as when 
0 the integrals involved are Gaussians or 
0 a recursive formula is available to carry out the successive integrations. 
The propagators for a free particle and for the harmonic oscillator of constant 
frequency fall in the first category. We quote here the Standard propagators: 
0 
0 
P(b 
For a free particle of mass m 
where T = t b - t a . 
For a free particle of mass m and frequency w - - 
1 = (27riti m w sin w T ) 3 1 2 exp{ 2;rwT ( [ X i + X i ] COS W T - 2Xb * Xa)} . 
[2] When a Feynman path integral is constructed, the action is rewritten, using 
the principle of canonical quantization, with the Substitutions 
where q is a generalized coordinate, and p is a conjugate momentum. 
23. Path Integrals 91 
[3] When L = L(x, x, t ) , the action functional in (23.7) can be discretized using 
the midpoint rule to obtain 
j=l 
For general L, the midpoint rule is required for the propagator as given 
by (23.8) to satisfy Schrödinger’s equation. However, in some cases, other 
formulas for the derivatives may be used, See Khandekar and Lawande [4]. 
When the classical Lagrangian has no explicit time dependence, the propa- 
gator can be written in the form 
[4] 
where the { E k } and the {&} are the complete Set of energy eigenvalues and 
eigenfunctions: H& = EkQ>k. This expansion is known as the Feynman-Kac 
expansion theorem. 
Note the oscillatory nature of the integral in (23.8) for small h. From 
the stationary phase technique (See Page 226) it is clear that the major 
contributor to the propagator Comes from paths for which 6s = 0 and 
b2S > 0. The equation 6s = 0 is Hamilton’s principle; its Solution is the 
classical path of the System. This equation is also equivalent to the usual 
Euler-Lagrange equations. If the integrand is expanded about the classical 
path, and terms out to second Order are kept, then the resultant propagator 
is equivalent to the WKB approximation of the Euler-Lagrange equations 
(See Zwillinger [ 111). 
Example l.B has the rudiments of a renormalization group calculation, See 
Zwillinger [ ii] . 
Other terms used for path integrals are Feynman path integrals and func- 
tional integrals. 
[5] 
[6] 
[7] 
R e ferences[l] 
[2] 
[3] 
S. Albeverio and R. Haegh-Krohn, Mathematical Theory of Feynman Path 
Integrals, Springer-Verlag, New York, 1976. 
R. P. Feynman and A. R. Hibbs, Quantum Mechanics und Path Integrals, 
McGraw-Hill Book Company, New York, 1965. 
T. Kaneko, S. Kawabata, and Y. Shimizu, “Automatic Generation of Feyn- 
man Graphs and Amplitudes in QED,” Comput. Physics Comm., 43, 1987, 
pages 279-295. 
D. C. Khandekar and S. V. Lawande, “Feynman Path Integrals: Some Exact 
Results and Applications,” Physics Reports, 137, No. 2-3, 1986, pages 115- 
229. 
P. K. MacKeown, “Evaluation of Feynman Path Integrals by Monte Carlo 
Methods,” Am. J. Phys., 53, No. 9, September 1985, pages 880-885. 
L. S. Schulman, Techniques und Applications of Path Integration, John Wiley 
& Sons, New York, 1981. 
[4] 
[5] 
[6] 
92 I1 Concepts and Definitions 
[7] R. G. Stuart and A. Ghgora-T., “Algebraic Reduction of One-Loop Feyn- 
man Diagrams to Scalar Integrals. 11,” Comput. Physics Comm., 56, 1990, 
pages 337-350. 
[8] G. J. van Oldenborgh, “FF - A Package to Evaluate One-Loop Feynman 
Diagrams,” Comput. Physics Comm., 66, 1991, pages 1-15. 
[9] F. W. Wiegel, Introduction to Path-Integral Methods in Physics and Polymer 
Science, World Scientific, Singapore, 1986. 
[ l O ] P. Zhang, “Simpson’s Rule of Discretized Feynman Path Integration,” J. Sci- 
entific Comput., 6, No. 1, March 1991, pages 57-60. 
[ll] D. Zwillinger, Handbook of Digerential Equations, Academic Press, New 
York, Second Edition, 1992. 
24. Principal Value Integrals 
Applicable to A formally divergent definite integral. 
Yields 
A convergent integral. 
Idea 
Many integrals are improper because of the limiting process involved. 
By restricting the way limits are taken in a definite integral, a convergent 
expression may sometimes be obtained. 
Procedure 
Suppose that f(z) has the Singular Point z in the interval ( a , b ) , and 
suppose that f(x) is integrable everywhere in the interval, except at the 
Point z. By definition (see Page 53), the integral s,” f(z) da: has the value 
where the limits are to be evaluated independently. If the value in (24.1) 
does not exist, but the value 
(24.2) 
does exist, then this value is the principal value, or the Cauchy principal 
value, of the improper integral I . This is denoted I = f, f(x)dz. Using 
the relation f,“ = f, +fbc, we can define the principal value of an integral 
with multiple singularities. 
b 
b 
24. Principal Value Integrals 93 
Example 
d x / ( x - 1). Because of the singularity at 
x = 1, this is an improper integral. Hence, I does not exist in the usual 
sense. However, I does have a principal value which is easy to determine: 
Consider the integral I = 
= lim {log 2) = log 2. 
7 4 0 
Notes 
[l] Suppose that f ( x ) is continuous on the interval [a ,b] , and vanishes only at 
the point z in this interval. If the first and second derivatives of f exist in a 
region containing the point z, and f'(z) # 0, then the integral 
is improper, but exists in a principal value sense. For the examplelpresented 
above, f(z) = x - 1 and f ' ( x ) = 1. 
The variables in a principal value integral may be changed under some mild 
restrictions. We have (See Davis and Rabinowitz [4], page 22): 
[2] 
Theorem: If z ( a ) = a and z(p) = b, and if, on the interval 
[a,p], z(<) is monotonic, z'(<) does not vanish, and z ( ( ) has 
a continuous second derivative, then 
where x = rn(t ) . 
[3] The integral s-", f ( x ) d x may be written in two forms: 
(AI s-", j(z) dz = limR,„ SORl j(x) da: + limR2+00 s,"" f(x) dx 
(B) s-", f(x) dx = l im~-+, (so, f ( X ) dx 4- st f ( X ) dX) - 
The integral s-", f(x) dx is convergent when the first Set of limits converge. 
If the second formulation converges, but not the first, then the integral 
is convergent in the sense of Cauchy and it is denoted f-", f ( x ) d x . For 
example, s-, x dx converges in the sense of Cauchy, but the integral is not 
convergent . 
Davies et al. [3] defines a higher-Order principal value to be given by: 
00 
[4] 
94 I1 Concepts and Definitions 
[5] Many methods for the numerical evaluation of principal value integrals 
may be found in the references. For example, in the work by Hunter and 
Smith [7], the location of the poles do not need to be known before the 
numerical algorithm is run. 
Mastroianni [ l O ] considers numerical methods for integrals of the form 
Rabinowitz [12] considers numerical methods for integrals of the form 
References 
G. Criscuolo and G. Mastroianni, “On the Uniform Convergence of Gaussian 
Quadrature Rules for Cauchy Principal Value Integrals,” Numer. Math. , 54, 
1989, pages 445-461. 
G. Criscuolo and G. Mastroianni, “On the Convergence of Product Formu- 
las for the Numerical Evaluation of Derivatives of Cauchy Principal Value 
Integrals,” SIAM J. Math. Anal., 25, No. 3, June 1988, pages 713-727. 
K. T. R. Davies, R. W. Davies, and G. D. White, “Dispersion Relations for 
Causa1 Green’s Functions: Derivations Using the Poincare-Bertrand Theo- 
rem and its Generalizations,” J. Math. Physics, 31, No. 6, June 1990, pages 
P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second 
Edition, Academic Press, Orlando, Florida, 1984, pages 11-15. 
G. A. Gazonas, “The Numerical Evaluation of Cauchy Principal Value Inte- 
grals via the Fast Fourier Transform,” Int. J. Comp. Math., 18, 1986, pages 
A. Gerasoulis, “Piecewise-Polynomial Quadratures for Cauchy Singular In- 
tegrals,’) SIAM J. Numer. Anal., 23, No. 4, 1986, pages 891-902. 
D. B. Hunter and H. V. Smith, “The Evaluation of Cauchy Principal Value 
Integrals Involving Unknown Poles,” BIT , 29, No. 3, 1989, pages 512-517. 
H. R. Kutt, “The Numerical Evaluation of Principal Value Integrals by 
Finite-Part Integration,” Numer. Math., 24, 1975, pages 205-210. 
J . N. Lyness, “The Euler-Maclaurin Expansion for the Cauchy Principal 
Value Integral,” Numer. Math., 46, No. 4, 1985, pages 611-622. 
G. Mastroianni, “On the Convergence of Product Formulas for the Evalua- 
tion of Certain Two-Dimensional Cauchy Principal Value Integrals,” Math. 
of Comp., 52, No. 185, January 1989, pages 95-101. 
P. Rabinowitz, “Numerical Evaluation of Cauchy Principal Value Integrals 
with Singular Integrands,” Math. of Comp., 55, No. 191, July 1990, pages 
P. Rabinowitz, “A Stable Gauss-Kronrod Algorithm for Cauchy Principal- 
Value Integrals,” Comput. Math. Appl. Part B , 12, No. 5-6, 1986, pages 
P. Theocaris, N. I. Ioakimidis, and J. G. Kazantzakis, “On the Numerical 
Evaluation of Two-Dimensional Principal Value Integrals,” Int. J. Num. 
Methods Eng., 15, 1980, pages 629-634. 
1356- 1373. 
2 77-2 88. 
265-276. 
1249-1 254. 
25. Transforms: To a Finite Interval 95 
25. Transforrns: To a Finite Interval 
Applicable to Integrals that have an infinite limit of integration. 
Yields 
An exact reformulation. 
Idea 
Many transformations map an integral on an infinite domain to an 
integral on a finite domain. Such a transformation may be needed before 
a numerical approximation scheme is used. 
Procedure 
Given the integral 
I = lm f(x) dx 
a numerical approximat ion t echnique of t he form 
N 
(25.1) 
(25.2) 
j=l 
may not be appropriate because of a fundamental indeterminacy in (25.1). 
If S is any positive number, then 
00 CO 
I = 1 f(x) d z = S i f(Sz) dx. 
Hence, the region where most of the “mass” of the integrand lies may have 
been (inadvertently) scaled to be outside of the range of the {xi}. Without 
some knowledge of the shape of f(x), it may be difficult to determine 
appropriate values for N and the { x i } in (25.2). 
One approach is to transform I to a finite domain and then use a 
numerical integration scheme on the finite domain. It is generally easier to 
understand the form of an integrand on a finite domain, than it is on an 
infinite domain. 
Under the transformation t = t ( x ) , we have I = Jl f [ x ( t ) ] Idx/dtl d t 
(see Page 109). Table 25 contains several usefultransformations of (25.1). 
96 I1 Concepts and Definitions 
Table 25. Several transformations of the integral &OO f(x) dx, with an infinite 
integration range, to an integral with a finite integration range. 
dx - Finite interval integral 
dt t ( 4 4 t ) 
e-" - logt 
1 
t 
-- 
t - X 
l + x 1 - t 
1 l + t 1 tanhx -log- - 
2 1 - t 1 - t2 
Notes 
[l] Another useful transformation, for any positive value of S, is: 
[2] The transformation lb f(t) dt = l:r$f (+) dt, valid for ab > 0, is useful 
in the two cases: 
0 b -, 00 with a > 0; 
0 a -+ -00 with b < 0. 
[3] If an integral has a known power-law singularity at an endpoint, then a 
transformation can be made to remove it. For example, suppose that f (x) 
diverges as (a: - u ) ~ near x = a (where -1 < (Y < 0). Then we can use 
where b > U . If, instead, f(x) diverges as ( b - x ) ~ near x = b (where 
-1 < ß < 0 ) , and b > U , then we can use 
[4] The Change of variables in this section can be performed analytically, or a 
numerical quadrature routine can perform the Change. 
26. Transforms: Multidimensional Integrals 97 
Re ference 
[ 11 W. Squire, Integration for Engineers und Scientists, American Elsevier Pub- 
lishing Company, New York, 1970, pages 174-176. 
26. Transforrns: 
Mult idirnensional Integrals 
Applicable to Definite multidimensional integrals. 
Yields 
A reformulation into a one-dimensional integral. 
Idea 
a single integral. 
Some classes of multidimensional integrals can be written in terms of 
Procedure 
involving a general function. 
written in terms of a single integral. 
The following list ing cont ains many mult idimensional integrals each 
These multidimensional integrals can be 
Two-Dimensional Integrals 
lT du l m f ’ ( p cosh 3: 4- q cos w sinh z) sinh z dz 
if p 2 > q2 and 1imZ+- f(z) = 0 (See Gradshteyn and Ryzhik [2], 4.620.1, 
Page 618). 
12T du lmf’ (pcoshz + (qcosw + T sinw) sinhz sinhz dz ) 
2~[signp] - - - I/= 
ifp2 > q2+r2 and lim„, f(z) = 0 (See Gradshteyn and Ryzhik [2], 4.620.2, 
Page 618). 
98 I1 Concepts and Definitions 
sin x sin2 y 
p - qcosx 
sin x sin y + r c o t y 
27r [sign p ] - - - 
d m 
ifp2 > q 2 + i 2 and limz+OO f (2) = 0 (see Gradshteyn and Ryzhik [2], 4.620.3, 
page 618). 
[I dx [I f' (p cosh x cosh y + q sinh x cosh y + r sinh y cosh y dy 1 
ifp2 > q2+r2 and limz+OO f (x) = 0 (Re Gradshteyn and Ryzhik [2], 4.620.4, 
Page 618). 
dx lT f (p cosh x + q cos w sinh x) sinh2 x sin w dw 
= 2 1 W f ([sign p] d m cosh x) sinh2 z dx 
if limz+OO f(x) = 0 (see Gradshteyn and Ryzhik [2], 4.620.5, page 619). 
00 1- f (u2x2 + b2y2) dx dy = -& 1 xf (z) dx 
if 1imzdm f(x) = 0 (see Gradshteyn and Ryzhik [2], 4.623, page 619). 
J 
if U, b, (Y, ß,p, q > 0 (See Prudnikov, Brychov, and Marichev [3], 3.1.2.1, page 
565). 
J Jf((Yx + py + 7) dx dy = 2 J_: I/Ef (td- + T) dt 
2 2 + y 2 < 1 
if U, b > 0 and (Y, ß, and y are real (See Prudnikov, Brychov, and Marichev [3], 
3.1.2.2, page 566). 
26. Transforms: Multidimensional Integrals 99 
(See Prudnikov, Brychov, and Marichev [3], 3.1.2.3, page 566). 
1' 1' f(xy)(l - x)"-'y"(l - Y ) ~ - ' dxdy = B(cr,P) J' f ( t ) ( l - t)U+P-' dt 
0 
if R e a > 0 and Reß > 0 (See Prudnikov, Brychov, and Marichev [3], 3.1.2.4, 
page 566). 
lmlm f (ax + by)e-Px-qY dx dy 
if a, b,p, q > 0 (See Prudnikov, Brychov, and Marichev [3], 3.1.3.1, page 567). 
if p, q > 0 (See Prudnikov, Brychov, and Marichev [3], 3.1.3.2, page 567). 
lw 1- f(xy)e-pz-q' dxdy = 2 1°0 KO (0 2 pqt f t dt 
if p, q > 0 (See Prudnikov, Brychov, and Marichev [3], 3.1.3.3, page 567). 
if p, q > 0 (see Prudnikov, Brychov, and Marichev [3], 3.1.3.4, page 567). 
Three-Dimensional Integrals 
lw dx JU" dw lnf (p cosh x + ( q cos w + T sin w ) sin I9 sinh x sinh2 x sin I9 d0 ) 
= 4 J l m f ([sign p] d m cosh x) sinh2 x dx 
ifp2 > q2+T2 and lim„, f(x) = 0 (See Gradshteyn and Ryzhik [2], 4.620.6, 
page 619). 
100 I1 Concepts and Definitions 
Lm d z L2% d u iT f (p cosh x + [(q cos w + r sin w ) sin 8 + s cosh 81 sinh x 
x sinh2 x sin 8 d8 
00 
= 4n Jo f ( [ s i g n p l ~ p ~ - q 2 - r 2 - 
if p2 > q2 + f 2 + s2 and limz+OO f (x) = 0 (See Gradshteyn and Ryzhik [2], 
4.620.7, page 619 or Prudnikov, Brychov, and Marichev [3], 3.2.3.4, page 
584). 
if a , b, c,p, q, r, Q, ß, -y > 0 and the integral on the right-hand side converges 
absolutely (See Prudnikov, Brychov, and Marichev [3], 3.2.2.1, page 583). 
xa-l ß - 1 7-1 y z f ( x + y + z ) d x d y d z 
if a , a , ß , y > 0 and the integral on the right-hand side converges absolutely 
(See Prudnikov, Brychov, and Marichev [3], 3.2.2.3, page 583). 
Multi-Dimensional Integrals 
This is known as the Dirichlet reduction (See Squire [6], pages 82-83). 
26. Transforms: Multidimensional Integrals 101 
where the Stars represent convolutions (See Sivazlian [ 5 ] ) . 
(See Prudnikov, Brychov, and Marichev [3], 3.3.1.1, page 585). 
.+--l) P n 
an d z 
2 1 2 0 2 2 2 0 ... xn 2 0 
where R is the region R : (:-ja' + ($"2 +. . .+ (:)On 
when the integral on the right converges absolutely (See Gradshteyn and 
Ryzhik [2], 4.635.1, page 620). 
( 2 1 2 0 2 2 2 0 ... xn 2 0 
where S is the region S : [ (:)"'+ (:)"'. ...+ ( $ 5 1 
when the integral on the right'converges absolutely, and the numbers qi, ai, 
and pi are positive. (See Gradshteyn and Ryzhik [2], 4.635.2, page 621). 
102 I1 Concepts and Definitions 
R 
XI 2 0 , 2 2 2 0 , S e . , xn 2 0 
X I + X ~ + . . .+ xn 5 1 , f (x) is con- where R is the region R : 
tinuous on (0 , l ) and qi 2 0 and T > 0 (See Gradshteyn and Ryzhik [2], 
4.637, page 622). 
if f(x) is continuous on the interval (0,R) (See Gradshteyn and Ryzhik [2], 
4.642, page 623). 
n dx p1 p1+p2 . . . xPl+P2+.. .+Pn-l X X 2 $3 
when the integral on the right converges absolutely (See Gradshteyn and 
Ryzhik [2], 4.643, page 623). 
- - 7r1T(n-1)’2 lT f (JpS + p ; + . . . + p z COS x) sinn-2 x dx 
r (q) 
if R is the region R = x: + x; + . . . + 5 1, n 2 3, and f (x) is continu- 
ous on the interval [-ß,ß], with ß = Jp; + p i + . . . + p i (See Gradshteyn 
and Ryzhik [2], 4.644, page 624). 
27. Transforms: Miscellaneous 103 
Notes 
[l] See also the section on how to Change variables on page 109. 
[2] Schwartz [4] considers a numerical technique for integrals of the form (nf=l s g i ( z i ) dxi) F fi(xi)). His technique is based on the use of 
integral t ransforms. 
References 
R. Y. Denis and R. A. Gustafson, “An S U ( N ) N Q-Beta Integral Trans- 
formation and Multiple Hypergeometric Series Identities,” SIAM J. Math. 
Anal., 23, No. 2, March 1992. 
I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, und Products, 
Academic Press, New York, 1980. 
A. P. Prudnikov, Yu. A. Brychov, and 0. I. Marichev, Integrals und Series, 
Volume 1, translated by N. M. Queen, Gordon and Breach, New York, 1990. 
C. Schwartz, “Numerical Integration in Many Dimensions. I,” J. Math. 
Physics, 26, No. 5, May 1985, pages 951-954. 
B. D. Sivazlian, “A Class of Multiple Integrals,” SIAM J . Math. Anal., 2, 
No. 1, February 1971, pages 72-75. 
W. Squire, Integration for Engineers und Scientists, American Elsevier Pub- 
lishing Company, New York, 1970. 
27. Transforrns: Miscellaneous 
Idea 
This section has a collection of transformations of integrals. 
Some Specific Forrns for 
One-Dimensional Integrals 
When an integrand has a specific form, there are some common trans- 
formations that might be useful. For example, if the integrand: 
(A) is a rational function of sinx and cosx, introduce U = tan(x/2) 
so that 
2u 1 - u2 2 du 
sinx = - cosx= - dx = - 
1 + u 2 ’ 1 + u 2 ’ 1 + U 2 ’ 
104 I1 Concepts and Definitions 
sin2 x 
dx becomes J C O S ~ x + cos x sin3 x For example, the integral J = 
du 
8u2(u2 + 1) J = J 
- I ) ( ~ ~ - 2u - 1)(u4 + 2u3 + 2u2 - 2u + 1) 
4u3 + 6u2 + 4u - 2 + 221 - 2 - - - - ] d u 1 1 = J [ u4 + 2u3 + 2u2 - 2u + 1 q U 2 - 2u - 1) + 1 - 1 
= 5 log + 2u3 + 2u2 - 2u + 1) + 5 log (qU2 -2u - I)) 
- log(u + 1) - log(u - 1) 
3 l ( 8u3 - q3 ) = -log 1 - 
= 6 log (1 + tan3 x) . 
is a rational function of sinhx and coshx, introduce U = tanh(x/2) so 
that 
221 1 + u 2 2 du sinhx = - coshx = 2, dx = - 
1 - U 2 , 1-U 1 - u2' 
is a rational function of x and d m , introduce x = cosv to get 
to case (A). 
is a rational function of x and J x T - , introduce x = coshv to 
get to case (B). 
is a rational function of x and d m , introduce U = x + d m 
so that 
is a rational function of x and Jux2 + bx + c, then the substitu- 
- . U - 
2a 
will reduce the integral to one of the above cases. 
is a rational function of x and U = /z, - then use the substi- 
tut ion 
b - U 2 d 
U = - x = --* 
2/cx+d a - u2c 
Integrals of the form S P ( a + bxn)Pdx, where rn, n and p are rational 
numbers, can be expressed in terms of elementary functions in each of the 
27. Transforms: Miscellaneous 105 
following cases (see Gradshteyn and Ryzhik [3] 2.202, Page 71): p is an 
integer; (rn + l ) /n is an integer; and p + (rn + l ) /n is an integer. 
Three useful transformations of definite integrals fiom Gradshteyn and 
Ryzhik [3] (3.032.1-3.032.3) are 
r / 2 
f(sinx)dx = 1 f(cosx)dx 
f (pcosx + qsinx) dx = 2 lr f ( ~ ~ c o s x ) dx 
f (sin 22) cos x dx = 1 f (COS' X) COS x dx. x / 2 
The Slobin transformation is (see Squire [6], Page 95) 
00 
f (t - i) dt = Jrn f ( t ) dt. 
-00 
The Wolstenholme transformation is (see Squire [6], Page 95) 
Two unnamed transformations for definite integrals (from Squire [6] Page 
94) are: 
Ir x f (sin x, cos2 x) dx = f (sinx, cos2 x) dx 
tan-l x 
f (x" + x - ~ > - dx = 
X X 
Dealing wit h Singularit ies 
Often, integrable singularities can cause a numerical routine to have 
trouble approximating an integral. While there are no universal principles 
to be applied, some simple techniques can be useful. 
Consider the integral I = 11% dx. Assuming that f(0) is finite, 
I has an integrable singu1arity.- "Bit a Computer routine evaluating this 
integrand near x = 0 may have trouble. 
I1 Concepts and Definitions 
If the f ( x ) term was not present, then we would note the indefinite 
integral J d x / & = 2&. Since the form fi appears, it might be 
reasonable to transform the integral to be on this scale. Changing 
variables via U = fi changes I to I = 2 Ji f ( u 2 ) d u . This integral no 
longer has a singularity at the origin. 
Another technique is to subtract out a singularity. For example, if 
f(0) has the finite value A , then I can be written as 
This new integrand has more smoothness than the original integrand, 
thus should be easier to evaluate numerically. 
Another technique is to break the integral up into several sub-integrals. 
For the above integral we might try 
where E is large enough so that a numerical routine does not have 
trouble with J1 , say E = 10-4. Then the integral J2 has a narrow 
region of integration; expansion methods, such as Taylor series, may 
yield a good numerical approximation. 
Of Course, commercial integration Software can automatically detect power 
singularities, and apply an appropriate (numerical) transformation to avoid 
comput at ional difficult ies. 
Products of Special Functions 
Piquette [5] considers integrals of the form I = J f ( x ) HEl R r j ( x ) dx, 
where R t ! ( x ) is of the i-th type of a Special function of Order p, obeying 
the set of recurrence relations 
(27.1) 
Here {U,, b,, c,, d,} are known functions. Most of the Special functions 
of physics, such as the Bessel functions, Legendre functions, Hermite func- 
tions, and Laguerre functions have recursion relations in the form of (27.1). 
The indefinite integral I is assumed to have the form 
1 1 1 m 
P l = o p Z = o p ,=o i=l 
27. Transforms: Miscellaneous 107 
Then the values of the {Ap} are determined, sometimes with the help of 
a differential equation. Piquette [5] obtains, for example, the following 
integral involving Legendre functions 
A Jacobian of Unity 
Sometime a judicious Change of variables in a multiple integral may 
make it easier to evaluate. Finding the correct Change of variables may 
sometimes be done by solving a partial differential equation. 
For example, changing variables in the usual way in two dimensions 
results in (see Page 109) 
(27.2) 
where F(u, U) := f (z(u, U), y (u , U)). If U is Chosen so that F(u, U) = F(u), 
and v is Chosen so that 
auav auav -1 1-1 = ä x & - a y z G - - 1, (27.3) 
then (27.2) becomes I = JJ F(u) du du, which may be easier to integrate. 
Note that equation (27.3) for v is a partial differential equation. 
Examde 
Suppose we have the integral I = 1'1' log (Y) dx dy. Choos- 
du du 
ing U = x2-y2 simplifies (27.3) to 2x-+2y- = 1 which has as a Solution ay ax 
v = + log@ + y). 
Using the U and v variables defined above, I becomes 
v3 (U) 
I = { Js% l;;:; + JI„ J1,, + Ji:" ll;:;} 1% dv du, 
where 
v1(u) = + log (3 + Js-u) 
v2(u) = a log (1 + d G i ) 
v3(u) = ;log (2 + d G ) 
v4(u) = a log (4 + JG). 
It is now straightforward to evaluate this integral. 
See Morris [4] or Squire [6] pages 12-13 for details on this method. 
108 I1 Concepts and Definitions 
Notes 
[l] If R(sin x, COS x) satisfies the relation R(sin x, COS z) = -R( - sin z, COS x) 
then it is normally convenient to make the Substitution z = cosx. This 
results in the formula 
JR(sinx,cosx)dx = - J R ( J i F Z , z ) - dz 
-Ji-..‘ 
[2] If R(sin x, COS x) satisfies the relation R(sinx, COS z) = -R(sinx, - COS x) 
then it is normally convenient to make the Substitution z = sinz. This 
results in the formula 
[3] If R(sin x, COS x) satisfies the relation R(sin x, COS x) = R( - sin x, - COS x) 
then it is normally convenient to make the Substitution z = tanx. This 
results in the formula 
R(sin x, COS x) dx = J 
For example, we find (compare to the first example in this section) 
sin2 x 
C O S ~ x + COS x sin3 x dx 
22 - 1 +‘-I dz 
= J [ 3 ( z 2 - z + 1 ) 3 ( 2 + 1 ) 
= 5 log (2 - z + 1) + 5 log (2 + 1) 
= 5 log (1 + 2) 
= 5 log (1 + tan3x) . 
References 
M. Abramowitz and I. A. Stegun, Hundbook of Muthemuticul Functions, 
National Bureau of Standards, Washington, DC, 1964 
M. S. Ashbaugh, “On Integrals of Combination of Solutions of Second-Order 
Differential Equations,” J. Phys. A : Muth. Gen., 19, 1986, pages 3701- 
3703. 
I. S. Gradshteyn and I. M. Ryzhik, Tubles of Integrals, Series, und Products, 
Academic Press, New York, 1980. 
W. L. Morris, “A New Method for the Evaluation of JJA f(x, y) dydx,” 
Amer. Muth. Monthly, 43, 1936, pages 358-362. 
J. C. Piquette, “Applications of a Technique for Evaluating Indefinite In- 
tegrals Containing Products of the Special Functions of Physics,” SIAM 
J. Muth. Anal., 20, No. 5, September 1989, pages 1260-1269. 
W. Squire, Integration ~ O T Engineers und Scientists, American Elsevier Pub- 
lishing Company, New York, 1970. 
I11 
Exact Analyt ical 
Methods 
28. Change of Variable 
Applicable to Definite integrals. 
Yields 
Idea 
Procedure 
A reformulation. 
A re-parameterization of an integral may make it easier to evaluate. 
Given the integral 
x=b 
I = s,=. f(X>dX, (28.1) 
if there exists a transformation of the form x = u ( y ) , which can be uniquely 
inverted on the interval U 5 x 5 b to give y = w(x), then 
I = 1 f ( u ( d ) = 1 f (v(Y)).il’(Y) &Y. (28.2) y=v(b ) y=v (b ) 
y=w(a) y=v(a> 
It is sometimes easier to evaluate (28.2) than to evaluate (28.1) directly. 
109 
110 I11 Exact Analytical Met hods 
X U 
Figure 28.1 Pictorial representation of a mapping from the {z,y} plane to the 
{ U , v} plane. 
Similar results hold in multiple dimensions. If there exists a one-to-one 
transformation from the region R in the xy-plane to the region S in the 
uv-plane, defined by the continuously differentiable functions x = z(u, v), 
y = y(u,v); and if the Jacobian 
(28.3) 
is nonzero in the region of interest, then 
R S 
(See Figure 28.1.) A listing of some common two-dimensional transforma-tions is in Table 28.1. 
In an orthogonal coordinate System, let {%} denote the unit vectors in 
each of the three coordinate directions, and let {u i } denote distance along 
each of these axes. The coordinate systern may be designated by the metr ic 
coef ic ients { 9 1 1 , g 2 2 , 9 3 3 } , defined bY 
(28.5) 
where {x1,22,23} represent rectangular Coordinates. Using the metric 
coefficients defined in (28.5), we define g = g11g22g33; the magnitude of 
the Jacobian of the transformation is then given by IJI = Js. Moon and 
Spencer [4] list the metric coefficients for 43 different orthogonal coordinate 
Systems. (These consist of 11 general Systems, 21 cylindrical Systems, and 
11 rotational Systems.) 
Operations with orthogonal coordinate Systems are sornetimes written 
in terms of the ( h i } functions, instead of the { g i i } terms. Here, hi = 6, 
so that IJI = Js = hlh2h3. For exarnple, cylindrical polar Coordinates are 
defined by x1 = r COS $, 22 = r sin 6, and 23 = z. Therefore, using {ul = r , 
212 = 8, u3 = z } , we find hl = 1, h2 = r , and h3 = 1. F’rom this we compute 
the magnitude of the Jacobian to be IJI = T . A listing of some common 
three-dimensional transformations, and their Jacobians, is in Table 28.2. 
28. Change of Variable 111 
Example 1 
Given t he integral 
(28.6) 
we choose to Change variables by x = u(y) = sin-l Y. This function can be 
uniquely inverted (On the range 0 5 x 5 $) via y = v(x) = sinx. Thus, 
This could also have been obtained directly by dy = v’(x)dx = cos dx. 
Our Change of variable then permits (28.6) to be written as 
y d y = - log(l+ Y Y 1 
$/=l 
--- - ycosx dy I = L0 1 + Y 2 C O S X l + y 2 
(28.7) 
Example 2 
Consider the integral I = sei x2 dx. If we choose to Change variables 
by y = v(x) = x2, then we note that v(x) cannot be uniquely inverted on 
the given interval of integration. Hence, we must break up the integral I 
into pieces, with ~ ( x ) uniquely invertible on each Piece. If we write I as 
1 
1 
I = I I + I2 = I1 x2dx + x2dx, 
then we can use the Change of variables {Y = x2, x = -Jy} on I1 and the 
Change of variables {y = x2, x = Jy} on 1 2 to obtain 
1 
&dy = (?y3li) 1 = - 1 I i = [ l x 2 d x = l O Y = = - dY 
0 3 2 3 
and 
We conclude that I = Il + I2 = + = 2 3 ‘ 
112 I11 Exact Analytical Methods 
U = 
Y ? u = 4 v = -2 
2 
Figure 28.2 The region of integration for Example 2, and the results of the 
coordinate transformation U = x + Y, = 2 - 2y. 
Example 3 
dx dy where 
R is the parallelogram shown in Figure 28.2. The sides of R are straight 
lines with equations of the form 
Suppose we wish to evaluate the integral I = s s (x + 
R 
x + y = c i , x - 2y = C a , 
with the values c1 = 1, c1 = 4, c2 = -2, and c2 = 1. If we introduce the 
new Coordinates 
then the region R corresponds to the rectangle 1 5 U 5 4, -2 5 v 5 1. 
This mapping is clearly one-to-one, and the Jacobian is given by 
u = x + y , v = x - 2 y 
1 - -- - 1 - Y> 1 
ab , v> a(u, v) 
a ( X 7 Y> 
J=-=-- 
3’ 1: -;I 
Therefore, we find 
I = 1 La(. + y)21Jl du du = l4 La du du = 21. 
Table 28.1 Some common two-dimensional transformations. 
128 
Notes 
I11 Exact Analytical Met hods 
The Version numbers of the Computer languages used above were Derive 
2.08, MACSYMA 417.100, MAPLE 5.0, Mathematica 2.1, REDUCE 3.4. 
All of the above Computer packages can perform Operations on integrals that 
the System cannot simplify. That is, if a System cannot simplify the integrals 
I = s” f(x) dx or J = Jab g(x) dx, it will still be able to differentiate I and 
numerically evaluate J . 
In DERIVE, presently, the only definite integrals that can be computed are 
those for which an indefinite integral can be computed. 
Sometimes a symbolic algebra System can determine a specific integral, yet 
it does not do so without coaching because the result is a mess. For example, 
the integral s dx/(x3 + x + 1) can clearly be integrated (the roots of a cubic 
can be found analytically, then partial fraction decomposition can be used). 
Yet Macsyma will not, unaided, carry out these steps. See Golden [3]. 
Packages that can handle a wider variety of integrals are constantly being 
created. The theory underlying the algorithms is described in the section 
on Liouville theory (See page 77). 
One must be wary when using Computer algebra packages, as errors can 
sometimes arise. For example, the integral I = s x k dx is problematic since 
there are two different forms for the answer depending on whether or not k 
is equal to -1. Some Systems will return xk+l/(k + 1) without any further 
information; others will ask the user whether or not k is equal to -1; still 
others will look to explicitly declared domains to determine if k could be 
equal to -1. 
If Derive cannot determine if k could be -1, then it returns the ex- 
Pression (xk+l - 1) /(k + 1) for the evaluation of I (See Stoutemyer [6]). 
This expression tends to logz as k + -1. Hence, this expression gives the 
correct result for any specific value of k, provided that a limit is used instead 
of simple Substitution. 
Most of the commercial Computer algebra packages also include a numeric 
integrator . 
Thanks are extended to Jeffrey Golden for running the test Suite in Mac- 
syma, Tony Hearn for running the test Suite in REDUCE, and Richard 
Pavelle for running the test Suite in MAPLE. 
References 
[l] B. W. Char, K. 0. Geddes, G. H. Gonnet, B. L. Leong, M. B. Monagan, 
and S. M. Watt, MAPLE V Library Reference Manual, Springer-Verlag, 
New York, 1991. 
DERIVE, Soft Warehouse, Inc., 3660 Waialae Avenue, Suite 304, Honolulu, 
HI 96816. 
J. P. Golden, “Messy Indefinite Integrals,” MACSYMA Newsletter, 7, No. 3, 
Symbolics, Inc., Burlington, MA, July 1990, pages 9-17. 
VAX UNIX MACSYMA Reference Manual, Symbolics Inc., Cambridge, MA, 
1985. 
G. Rayna, R E D UCE: Software for Algebraic Computation, Springer-Verlag, 
New York, 1987. 
[2] 
[3] 
[4] 
[5] 
138 I11 Exact Analytical Methods 
I 
Figure 30.8 Bromwich contour used in inverse Laplace transforms. 
Figure 30.8. With a closed contour, of Course, Cauchy’s theorem can be 
used. 
Consider the inverse Laplace transform of the function F ( s ) = l/(s2 + 
4). This function has singularities at s = f22 , so we must take U > 0. 
Defining a new integral with the Bromwich contour 
we note that the only poles are at s = f 2 i . Therefore, the residue theorem 
yields 
(residue at s = 2i) + (residue at s = -22) = - sin2t. I : 
The difference between f ( t ) (what we Want) and K ( t ) (what we have 
calculated) is the integral scR. As we now Show, this integral vanishes 
asR-00. 
Let 2 = Res along CR. Then, for positive t , we have (e ts ( < etx < eta 
along CR. Hence, 
since U is fixed. We obtain the final result f ( t ) = sin 2 t . 
30. Contour Integration 
Notes 
139 
Jordan’s lemma states: If C R is the contour z = Reie, 0 5 8 5 7r, then sc, leiZl ldzl < 7r (See Levinson and Redheffer [7]). This is often useful for 
estimat ing int egrals. 
The integral in Example 1 had the form I = /:* f(cos 8, sin 8) de . Integrals 
of this form can be recast as contour integrals by Setting z = eie. As 8 varies 
from 0 to 27r, z traverses a unit circle C counterclockwise. Hence, 
Loop integrals are contour integrals in which the path of integration is 
given by a closed loop. For example, Hankel’s representation of the Gamma 
function when Rex > 0 is given by 
where the contour C lies in the complex plane cut along the positive real axis, 
starting at 00, going around the origin once counterclockwise, and ending 
at 00 again. 
In the above theorems, the contours were sometimes described as being 
“traversed in the positive sense.” Consider a closed region surrounding t he 
contour C ( t ) . Consider the vector giving the direction of increasing t ; if a 
counterclockwise rotation of n/2 causes the vector to Point into the interior 
of the closed region,then the contour C is being traversed in the positive 
sense and the region is positively oriented. 
Plaisted [9] Shows that the following Problem involving the evaluation of a 
contour integral is NP-hard: 
Given integers b and N and a Set of k sparse polynomials 
with integer coefficients {pi(z)}, it is NP-hard to determine 
whether the following contoui integral is zero: 
where C is any contour including the origin in its interior. 
See also Garey and Johnson [5]. 
Gluchoff [6] presents the following interpretation of a contour integral: 1 f ( z ) d z = L ( C ) av [f(4T(z)l (30.19) 
where L ( C ) is the length of C , T ( z ) is the unit tangent to C at z , and “av” 
denotes the averaging function. 
For example, for the integral I = h z , = R d ~ / z we identify L ( C ) = 27rR 
and T ( z ) = iz/R so that (30.19) becomes 
ZEC 
d z - = 27rR av [tg] = 27ri av [l] = 2ni. 
J z l = R Izl=R lzl=R 
140 I11 Exact Analytical Methods 
[7] 
[8] 
Several of the references describe quadrature rules for numerically approxi- 
mating a contour integral. 
The numerical integrator in the language Mathematica allows an arbitrary 
path in the complex place to be specified. For example, consider the integral 
in (30.5). The integration contour may be deformed to the Square with 
vertices at {&l,&i} without changing the value of the integral. If Mathe- 
matica is input “NIntegrate[-2I/(zn2+4z+1) , z , l , I , -1 , - I , l ]” , then 
the result is 3.6276 + 0. I, which is the correct numerical value. (Here 
I is used to represent i.) 
The theory of multidimensional residues is based on the general Stokes 
formula and its corollary, the Cauchy-Poincare integral theorem. Serious 
topological difficulties arise for analytic functions of several complex vari- 
ables because the role of the Singular Point is now played by surfaces in C”. 
These surfaces generally have a complicated structure requiring tools such 
as algebraic topology. 
[9] 
References 
[l] 
[2] 
[3] 
B. Braden, “Polya’s Geometric Picture of Complex Contour Integrals,” 
Math. Mag., 60, No. 5, 1987, pages 321-327. 
P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second 
Edition, Academic Press, Orlando, Florida, 1984, pages 168-171. 
D. Elliot and J. D. Donaldson, “On Quadrature rules for Ordinary and 
Cauchy Principal Value Integrals over Contours,” SIAM J. Numer. Anal., 
14, No. 6, December 1977, pages 1078-1087. 
R. J. Fornaro, “Numerical Evaluation of Integrals Around Simple Closed 
Contours,” SIAM J. Numer. Anal., 1, No. 4, 1973, pages 623-634. 
M. R. Garey and D. S. Johnson, Computers und Intractability, W. H. Free- 
man and Co., New York, 1979, page 252. 
A. Gluchoff, “A Simple Interpretation of the Complex Contour Integral,” 
Amer. Math. Monthly, 98, No. 7, August-September 1991, pages 641-644. 
N. Levinson and R. M. Redheffer, Complex Variables, Holden-Day, Inc., San 
F’rancisco, 1970, Chapter 5, pages 259-332. 
J. N. Lyness and L. M. Delves, “On Numerical Integration Round a Closed 
Contour,” Math. of Comp., 21, 1967, pages 561-577. 
D. A. Plaisted, “Some Polynomial and Integer Divisibility Problems Are 
NP-Hard,” in Proc. 17th Ann. Symp. on Found. of Comp. Sci., IEEE, Long 
Beach, CA, 1976, pages 264-267. 
[4] 
[5] 
[6] 
[7] 
[8] 
[9] 
31. Convolution Techniques 
Applicable to 
of terms. 
Some one dimensional integrals that contain a product 
Yields 
An exact evaluation in terms of other integrals. 
31. Convolut ion Techniques 141 
Idea 
of the integral may be determined by a sequence of integrals. 
Procedure 
Given the functions f(z) and g(z), and an integral Operator I[], define 
the functions F ( [ ) = I[f(x)] and G([) = I [g(z )] . For many common 
integral Operators there is a relation that relates an integral of f and g, of 
a specific form, to an integral involving F and G. These relations are often 
of the form 
If the original integral can be written as a convolution, then the value 
where A[] is an integral Operator. Such a relation is known as a convolution 
t heorem. 
Convolution theorems may sometimes be used to simplify integrals. 
Given an integral in the form of the right-hand side of (31.1), it may be 
easier to evaluate the left-hand side of (31.1). In the following examples, 
upper case letters denote transforms of lower case letters. 
Example 1 
For the Mellin transform, defined by 
t he convolution t heorem is 
00 c+im 1 f (E) g(u)$ = 23-i 1 c-aOO F(s)G(s)z- 'ds 
where the integral on the right-hand side is a Bromwich integral. 
Example 2 
For the Laplace transform pair, defined by 
F ( s ) = L[f(t)] = J" e--"f(t) d t , 
0 
the convolution theorem is 
r t 
f ( r ) g ( t - 7) d r = L-' [F(s)G(s)] . 
J o 
142 I11 Exact Analytical Methods 
Example 3 
For the Fourier transform pair, defined by 
F ( t ) = .F[f(z)] = I J e - i t 2 f ( z ) d z , J27; -00 
f(z) = P [ F ( t ) ] = - J eit2F(t)cit, 6 -00 
the convolution t heorem is 
Notes 
[l] Bouwkamp [2] uses the convolution theorem for Fourier transforms to Show 
t hat 
2n = - [l- c~DKo ($(YD) Ii ($(YD)]. dr d$ 12T dr2 -t D2 - ~ ~ D c o s ~ 
References 
[l] A. Apelblat, “Repeating Use of Integral Transforms-A New Method for 
Evaluation of Some Infinite Integrals,” IMA J. Appl . Mathematics, 27, 1981, 
pages 481-496. 
C. J. Bouwkamp, “A Double Integral,’’ Problem 71-23 in SIAM Review, 14, 
No. 3, July 1972, pages 505-506. 
E. Butkov, Mathematical Physics, Addison-Wesley Publishing Co., Reading, 
MA, 1968. 
N. T. Khai and S. B. Yakubovich, “Some Two-Dimensional Integral Trans- 
formations of Convolution Type,’’ Dokl. Akad. Nauk BSSR, 34, No. 5, 1990, 
pages 396-398. 
[2] 
[3] 
141 
32. Differentiation and Integration 
Applicable to 
Yields 
Definite and indefinite integrals. 
An alternative representation of the integral. 
Idea 
By differentiating or integrating an integral with respect to a param- 
eter, the integral may be more tractable. This Parameter may have to be 
introduced into the original integral. 
32. Differentiation and Integration 143 
Procedure 
Given an integral, try to differentiate or integrate that integral with 
respect to a Parameter appearing in the integral. If there are no Parameters 
appearing in the integral, insert one and then perform the differentiation 
or integration. 
After the integration is performed then either an integration or a 
differentiat ion remains. 
Example 1 
eFt2 COS zt d t , differentiation Shows that 
u(z) satisfies the differential equation U' + $zu = 0. (This was obtained 
by integration by Parts.) This differential equation is separable (see Zwill- 
inger [3]) and the Solution is found to be u(x ) = Ce-x2/4, for some con- 
stant C. From the defining integral, u(0) = so e d t = ifi, so we have 
the final answer 
(32.1) 
Given the integral u(z) = 
00 - t 2 
lm e-t2 cos xt d t = i f i e - x 2 / 4 . 
Example 2 
Given the integral 
I ( p ) = LW zsin(z)e-px2 d z 
J ( u , p ) = lrn z sin(uz)e-px2 d z . 
we might introduce a Parameter U and consider instead 
(32.2) 
Note that I ( p ) = J(1,p). Both sides of (32.2) may be integrated with 
respect to U to determine 
/ a J ( u , p ) d u = I a Lrn 5 sin(uz)e-px2 d z d u 
(32.3. u-b) 
= - LW cos(uz)e-px2 d z . 
This last integral may be evaluated by a simple Change of variable and 
using (32.1). The result is 
Differentiating (32.4) with respect to U results in 
and so 
(32.4) 
144 I11 Exact Analytical Methods 
Example 3 
Given t he int egal 
(32.5) 
d 
db 
we differentiate with respect to b to obtain -K(b, q ) = 
which is the Same as (32.3.b). Therefore 
cos(bx)e-qX2, 
(32.6) 
F'rom (32.5) we note that K(0,q) = 0. 
differential equation in (32.6) with this initial condition. The Solution is 
Hence, we need to solve the 
(32.7. U-C) 
= - 7r erf (&) 
2 
where we have recognized (32.7.b) as being an error function (see Page 172). 
Notes 
[l] As another example of the technique illustrated in Example 3, consider the 
integral(32.8) 
Differentiating with respect to y produces 
dx 
1 + s y 2 1 + z 2 
1 (32.9) 
Ir -- - 
l + Y 
if y > 0. Rom (32.8) we recognize that F ( 0 ) = 0. Therefore, we can 
integrate (32.9) to determine 
[2] Squire [2] Starts with the identity so1 xt-' dt = t - l , and integrates with 
respect to t (from t = 1 to t = U ) to obtain 
33. Dilogarit hms 145 
[3] Many other clever manipulations based on integration and differentiation 
can be conceived. For example (again from Squire [2]), the identity 2-l = 
JOm e-xt dt can be used in the following way: 
References 
[l] 
[2] 
[3] 
J. Mathews and R. L. Walker, Mathematical Methods of Physics, W. A. 
Benjamin, Inc., NY, 1965, pages 58-59. 
W. Squire, Integration for Engineers und Scientists, American Elsevier Pub- 
lishing Company, New York, 1970, pages 84-85. 
D. Zwillinger, Handbook of Diflerential Equations, Academic Press, New 
York, Second Edition, 1992. 
33. Dilogarithms 
Applicable to Integrals of a Special form. 
Yields 
An exact Solution in terms of dilogarithms. 
Idea 
Integrals of the form J” P(x , a) logQ(z, a) dx, where P( , ) and 
Q( , ) are rational functions and R = A2 + Bx + Cx2, can be transformed 
to a canonical form. 
Procedure 
Given the integral 
I = 1” P(x; fi) log Q(x, I&) dx, (33.1) 
make the Change of variable = A + xt. This transformation results in 
d ~ 2 + BX + C X ~ - A t = 
X 
2At - B 
c - t2 X = 
dx = [AC - Bt + At2] d t 
(C - t2 )2 
fi - AC-Bt+At2 - c - t2 
146 I11 Exact Analytical Methods 
so that (33.1) becomes 
(33.2) 
where P and 
expansion can be used on P to obtain 
are rational functions of t . In principle, partial fraction 
a m n - P = P(t ) + 
m,n (ßmn + t)" 
(33.3) 
where P(t ) is a polynomial in t. Using (33.3) in (33.2), and expanding, leads 
to many integrals. Those integrals that have the form JP(t)log(y + t ) d t 
log(y + t ) d t , for n # -1, can be evaluated by or the form 
integrating by Parts. The only integral that cannot be evaluated in this 
manner is 
J (LT t > n 
J lo",';; t ) dt = { log(a - y> log (Y) - Li2 (s) + c, if <r # y, 
if CY = y, 
where Li2 ( ) is the dilogarithm function and C is an arbitrary constant. 
The dilogarithm function is defined by 
f log2(y + t ) + c, 
(33.4) 
log(1 - 2) 
~ i 2 (x) = - 1 dx. 
X 
(33.5) 
Hence, the integral in (33.1) can always be integrated analytically 
A few integrals in terms of the dilogarithm and elementary functions. 
involving dilogarithms are shown in Table 33. 
Example 
The integral 
I = J" log ( x + d=) dx (33.6) 
becomes under the Change of variable & = d m ' = 1 + xt (which is 
the Same as x = 2t/(l - t2)) 
XdGZ 
[log(l + t ) - log(1 - t )] d t , 
= Li2 ( t ) - Li2 (-t) , (33.7) 
33. Dilogarithms 147 
Table 33. Some integrals involving dilogarit hms. 
1 
4 + - log2(1 + x2) - log(1 - x) log(1 + x2) 
7r2 i k ( l + u ) + y xa-lLi2 (2) dx = - - 
6u 1' U 2 
Notes 
[l] 
[2] 
Tables of the dilogarithm may be found in Lewin [2]. 
An extension of the dilogarithm function of one argument is the dilogarithm 
function of two arguments: 
where, of Course, Li:! (x,O) = Li2 (z) for -1 5 x 5 1. The higher Order 
logarithmic functions are defined by Li,(x) = sox Lin-l(x)/x dx. All of these 
functions have been well studied; recurrence relations and other formulae 
have been determined for each (See Lewin [2].) 
References 
[l] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 
National Bureau of Standards, Washington, DC, 1964, Section 27.7, page 
1004. 
L. Lewin, Dilogarithms und Associated Functions, MacDonald & Co., Lon- 
don, 1958. 
L. Lewin, Polylogarithms and Associated Functions, North-Holland Publish- 
ing Co., New York, 1981. 
[2] 
[3] 
148 I11 Exact Analytical Methods 
34. Elliptic Integrals 
Applicable to 
Yields 
An exact Solution in terms of elliptic functions. 
Idea 
Integrals of the form Jz R(x, JTT.>) da: where R( , ) is a rational func- 
tion of its arguments and T(x) is a third of fourth Order polynomial can be 
transformed to a canonical form. 
Integrals of a Special form. 
Procedure 
Given an integral of the form 
(34.1) 
T = u4x4 + u3x3 + u2x2 + als: + uo, and each fi(x) represents a polynomial 
in x, rewrite the integrand as 
(34.2) 
where Rl(x) and R2(z) are rational functions of x. Clearly, the integral 
J Rl(x) da: can be evaluated in terms of logarithms and arc-tangents (see 
Page 183). 
We can always write T in the form (See Whittaker and Watson [ lO] ) 
T = [Al(a: - a)2 + Bl(x - p)2] [ A ~ ( x - C Y ) ~ + &(a: - p)’]. Then, chang- 
ing variables by t = (x - a) / (x - p) results in 
where S := 1/(A1t2 + BI) (A2t2 + B2). Hence, we find 
34. Elliptic Integrals 149 
where R3 is a rational function. We can now write R3(t) = R4(t2) + 
tR5(t2), where R4 and R5 are also rational functions (by, say, R4(t2) 
(R3(t) + R3(-t))/2 and R5(t2) (R3(t) - R3(-t))/2t). Clearly, the in- 
tegral S t R 4 ( t 2 ) / a d t can be evaluated in terms of logarithms and arc- 
tangents by introducing the variable z = t2 (see Page 183). By a partial 
fraction decomposition, we can write 
t2s 1 -- R4(t2) - c O s - + E 
Js S Js i , S (i + nit2) 'Js 
(34.3) 
for some constants {aS}, {P i+} , and {ni}. By integrations by Parts, it turns 
out that we only need to be able to evaluate integrals of three canonical 
forms to determine (34.3) completely. Thus, knowledge of these canonical 
forms allow us to determine (34.1) completely. These three canonical forms 
are, in Legendre's notation: 
0 Elliptic integrals of the first kind (Iyanaga and Kawada [8], Page 1452): 
dt sin 4 . (34.4) 
= 1 J(1- P)(l - P t 2 ) 
0 Elliptic integrals of the second kind (Iyanaga and Kawada [8], Page 
1452): 
0 Elliptic integrals of the third kind (Iyanaga and Kawada [8], Page 
1452): 
(34.6) 
150 I11 Exact Analytical Methods 
0 t her Manipulations 
There are many ways in which to manipulate an integral to obtain 
an elliptic integral of the first, second, or third kinds. For example, by a 
partial fraction decomposition of (34.2), we can write 
X 8 1 -- R2(x) - E As- + E Bi„ 
& 8 fi z , s ( x - CJ&' 
for some constants {A8}, {Bi+}, and {Ci}. By writing 
T = a4x4 + a3x3 + a2x2 + alx + ao 
(34.7) 
= b 4 ( X - c)4 + b3(x - c)3 + b2(x - c)2 + b l ( X - c) + bo, 
we See that we only need to be able to evaluate integrals of the form 
xs dx 
I8 = J z d x and J, = J X ( , (34.8) 
- c y f i 
in Order to evaluate (34.1). The recurrence relations 
(S + 2)a41s+3 + +3(2s + 3)18+2 4- a2(s 4- l)Is+l + Sal(2s + l)Is 
+ saoISg-1 = x s J r 
- sboJs+l = ( x - c)'@ 
for s = 0, 1 ,2 , . . . 
(2 - S)b4J~-3 + $b3(3 - 2S)J8-2 -I- b2(1 - S)J8-i Sba(1 - 2S)J8 
for s = 1 ,2 ,3 , . . . 
(34.9) 
allow some manipulation of the {In} and the {Jn}. 
the exact form of T ( x ) . For example, if T ( x ) = Q ~ ( x ) & ~ ( x ) , where 
There are several different methods for calculating Io, depending on 
Q l ( x ) = ax2 + bx + c, 
&2(x) = dx2 + ex + f, (34.10) 
then the Change of variable z = , / Q ~ ( x ) / & ~ ( x ) allows I0 to be written m 
dx 
dmam 
= *J' dz 
d(e2 - 4df)z4 + 2(2af + 2cd - be)z2 + (b2 - 4ac) 
(34.11.~-b) 
Note that the radical in (34.11.b) is the discriminant of &2z2 - &I (i.e., 
what would appear in the radical if the quadratic formula were used to 
solve &2z2 - Q1 = 0 for the variable x ) . 
34. Elliptic Integrals 151 
Example 1 
Suppose we have the integral 
dx I = J x 
J"(3x2 + 22 + 1) * 
Since this has the form of (34.10), with Ql (x ) = x and Q2(x) = 3x2+2x+1 
we make the Change of variable 
This leads to (using (34.11)) I = & 2 / 4 1 - 4t2 - 8t4 dt. Defining b2 = 
(1 + &)/4 and -a2 = (1 - &)/4, we observe that I can be written in the 
form 
(34.12) 
This Standard form can be found in Abramowitz and Stegun [l], 17.4.49. If 
the lower limit on the integral in (34.12) is taken to be b, then the integral 
is equal to 
where nc is a Jacobian elliptic function (See Table 34). 
Example 2 
Given the integraldx 
I = LX J(5x2 - 42 - 1) (12x2 - 4x - 1)' 
we Change variables by y = l /x to obtain 
-dY 1/x 
I = L J(5 - 4y - y2) (12 - 49 - y2) * 
The Change of variable t = ;(Y + 2) results in 
dt (1+22)/32 I = - - i L J-' 
= -I [. (7, 1 + 2 x 4) 3 - F (1, a)] . 
4 
152 I11 Exact Analytical Methods 
Example 3 
Given the integral 
dx 2 
I = 1 J(22 - 22) (3x2 + 4) ' 
we Change variables by t = ( 2 - 32) / (6 + 32) to obtain 
I 
I = ~ J ~ dt 
0 J(1- 9t2) (1 + 3t2)' 
Introducing v 2 = 1 - 9t2, we find 
Table 34. Some relationships between elliptic integrals, elliptic functions in 
Legendre notation, and Jacobian elliptic functions (from Abramowitz and Ste- 
gun fl], Page 566). In the following, x = sin4. 
If U > b and k2 = (u2 - b 2 ) / u 2 , then 
d t = sc-l (:I k 2 ) 
(t2 + u2)(t2 + b2) F ( 4 , k ) = u i x 
X 
with 4 defined by tan 4 = -. b' 
d t = CS-' (%I k 2 ) 
(t2 + u2)( t2 + b2) 
U with 4 defined by tan4 = ;; 
.Ir 
d t = nd-' (51 k 2 ) 
(u2 - t2)(t2 - b2) 
u2(x2 - b 2 ) . 
x2(u2 - b2) ' with 4 defined by sin2 4 = 
d t = dn-' (EI k 2 ) 
(u2 - t2)(t2 - b2) . . 
u2 - x 2 with 4 defined by sin2 4 = - 
a2 - b2 ' 
34. Elliptic Integrals 153 
If a > b and k2 = b2 /a2 , then 
dt ' 
= sn-' (51 k 2 ) 
J(u2 - t2)(b2 - t 2 ) 
with 4 defined by sin 4 = 2. 
b ' 
dt = cd-' (:I k 2 ) b 
(a2 - t2)(b2 - t 2 ) 
F(4'k) = a l 
a2(b2 - x 2 ) . 
b2(a2 - x 2 ) ' 
with 4 defined by sin2 4 = 
dt = dc-' (EI k 2 ) 
( t2 - a2)(t2 - b 2 ) 
x 2 - a2 with 4 defined by sin2 4 = 7- 
x - b 2 ' 
dt = ns-' (%I k 2 ) 
( t2 - a2) ( t2 - b2) 
with 4 defined by sin 4 = 2. 
X 
~~ ~~ ~ 
If k2 = u2/(u2 + b 2 ) , then 
= nc-' (:I k 2 ) F ( 4 , k ) = d-1' dt 
J(t2 + a2)(t2 - b2) 
b with 4 defined by COS 4 = -; 
.1: - 
F ( 4 , k) = J- Jm dt = ds-' ( I k 2 ) 
J(t2 + a2)(t2 - b2) JW 
a2 + b2 with 4 defined by sin2 4 = - 
a2 + x 2 * 
If k2 = b 2 / ( a 2 + b 2 ) , then 
I . 
F ( 4 , k) = d m L x dt =sd-' ( X J . 2 + b . ab I k 2 ) 
J(t2 + a2)(b2 - t2) 
x2((a2 + b 2 ) . 
b2(a2 + z2) ' with 4 defined by sin2 4 = 
= cn-' (51 k 2 ) F ( 4 , k) = Jol+bZJb dt 
J(t2 + a2)(b2 - t 2 ) 
X with 4 defined by COS 4 = - 
b ' 
154 111 Exact Analytical Methods 
Notes 
[l] For the elliptic integrals in Legendre’s notation, k is called the Parameter 
or modulus, k’ = 4- is called the complementary modulus, 4 is called 
the amplztude, and for k = sina we define a to be the modular angle. In 
other representations of the elliptic functions, the variables m = k2 and 
m1 = 1 - m = kI2 = 1 - k2 are sometimes used. 
The elliptic integrals of the first and second kind are said to be complete 
when the amplitude is 4 = 7r/2, and so x = 1. The following Special notation 
is then used: 
[2] 
E = E ( k ) = E (:, k) = J C F Z G d O . 
0 
The complementary values are defined by K’ = K’(k) = K(k‘) = K ( d m ) 
and E’ = E’(k) = E(k’) = E ( 4 n ) . These values are related by 
Legendre’s relation: EK’ + E’K - KK’ = 7r/2. 
The 9 Jacobian elliptic functions, {cd, cs, dc, ds, nc, nd, ns, sc, sd}, can be 
defined in terms of the three “basic” Jacobian elliptic functions: sn, cn, and 
dn. We have the Standard relationships (from Abramowitz and Stegun [l], 
16.3) 
[3] 
cn u dn u 1 
c d u = - d c u = - n s u = - 
dnu’ cnu’ snu’ 
dn u d s u = - sn u 1 d n u ’ cnu’ snu ’ 
sn u cn u 
d n u ’ cnu’ sn u 
s d u = - n c u = - 
n d u = - s c u = - csu = -. 1 
The “basic” functions may be calculated from 
snu = sn(ulk2) = sin4 
cnu = cn(ulk2) = cos4 = JZZG 
dnu = dn(uIk2) = J- 
(34.13) 
where U is determined by an elliptic function of the first kind: U = F ( 4 , k). 
(That is, sn is the inverse function to F . Observe: sn(F(4,k)lk2) = sin4.) 
Tables of the Jacobian elliptic functions may be found in Abramowitz and 
Stegun [l]. A superscript of -1 on any one of these twelve functions denotes 
the inverse function. 
Table 34 relates the Legendre elliptic functions to the Jacobian elliptic 
functions. 
34. Elliptic Integrals 155 
[4] The differential equation (Y’)~ = 1 - y2 with y(0) = 0 has the Solution 
y(x) = sin(x), with a period of 
Analogously, the differential equation (Y’)~ = (l-y2)(1-Ic2y2) with y(0) = 0 
has the Solution y(x) = sn(x) = sn(xlIc2), with a period of 
dx P = 4J11 = 4K(k). Jc 1 - x2)( 1 - I c 2 2 2 ) 
[5] The Jacobian elliptic function sn(u) satisfies (See (34.13)) 
Under the transformation t = sine and x = sinq5 this becomes (See (34.4)) 
dt I k 2 ) = x . 
(1 - t2)( 1 - P t 2 ) 
Since sn(zl0) = sinz, the limit of Ic -, 0 produces 
sin (1’ L) = x or sin (sin-1 x> = x. 
J C F 
[6] Technically, any doubly periodic nieromorphic function is called an elliptic 
function. All of the usual elliptic functions are doubly periodic; for example, 
sn x = sn(x + 2iK’) = sn(x + 4K). 
Carlson [4]-[7] has introduced a new notation for the elliptic functions that 
preserves certain symmetries. Define four functions: 
[7] 
dt 00 
RF(Z,Y, 2) = - 1 d ( t + x)(t + y)(t + 2) 
dt 00 
RJ(x, z, w, = : JI J(t + x)(t + y)(t + z)(t + w) 
Each function has the value unity when all of its arguments are unity, and Rc 
and RJ are interpreted as Cauchy principal values when the last argument 
is negative. These functions may be calculated by the numerical routines 
presented in Carlson [5]-[6]. 
156 I11 Exact Analytical Methods 
The relationship between Legendre's notation and Carlson's notation 
is as follows: 
F ( 4 , k ) = sin 4Rp(cos2 4 , l - k2 sin2 4, 1) 
E($, ) = sin #Rp(cos2 4, 1 - k2 sin2 4, 1) 
- $ k2 sin3 ~ R D (cos2 4 , l - k2 sin2 4, 1) 
+ $n sin3 ~ R J (cos2 4, 1 - k2 sin2 4 , 1 , 1 - n sin2 4) 
lI(4, n, k ) = sin 4Rp(cos2 4, 1 - k2 sin2 4, 1) 
From the above we find E ( k ) = Rp(0 , 1 - k 2 , 1) - $ k 2 R D ( 0 , 1 - k 2 , 1) and 
K ( k ) = RF(O, 1 - k 2 , 1). Use of Carlson's notation for elliptical integrals 
dramatically reduces the number of separate cases that need to be tabulated 
in integral tables. 
The Weierstrass P function is an elliptic function defined by [8] 
1 1 1 2 (w-2) P(u) = P(u;w1,w2) = - + u2 n= .... -1.0.1 .... 
m= ...> -l,O,l,... 
n=2mw1+2nw2 
where the prime means that the term n = m = 0 is not present in the sum. 
The function z = P(u) is the inverse function of the elliptic integral (See 
Iyanaga and Kawada [8], page 483) 
(34.14) 
Weierstrass showed that any elliptic integral could be transformed into terms 
of the form of (34.14) by a linear fractional transformation. 
It is not hard to See that an integral of the form I = s Rl(z, dm-) dx, 
where R1 is a rational function and +(x) is a quartic polynomial, can be 
mapped to the form of (34.14). With a bilinear transformation, x = - 
dz = 
d 4 z 3 - g2.z - Q 3 . 
[9] 
CrY +ß 
YY + 6 
(with I 5 I # 0), I takes the form I = sRz(y, d m ) dy, where R2 is 
a rational function and +(Y) is a quartic polynomial. By an appropriate 
choice of coefficients in the transformation, + can have the form +(Y) = 
4Y3 - 92Y - 93. 
To understand this transformation, let c be a root of 4 ( x ) , and presume 
that +(z) does not have multiple roots. Then the transformation x = c+ l/z 
results in 
4'(c)z3 + +4"(c)z2 + . . . 
2 4(4 = 
That is, it yields a new elliptic integral that only involves a cubic polynomial. 
If we further let z = ay + b, then, for a suitable choice of b the coefficient of 
y2 in the cubic polynomial will vanish. Finally, by choice of U we can force 
the coefficient of y3 to be 4. For more details, See Akhiezer [2], pages 15-16. 
F’rullanian Integrals 157 
Integrals of the form I = s R ( x , d m ) dx, where R is a rational function 
and E is polynomial of degree n > 4 are called hyperelliptic integrals. If 
n = 2p + 2 (where p is an integer) then, by rational transformations, one 
can obtain an equivalent integral in which the polynomial is of degree 2p+ 1 
(See the previous note). See Byrd and Friedman [3] for details. 
Consider an ellipse described by the parametric equations x = acose andy = bsin8, with b > U > 0. The length of arc from 8 = 0 to 8 = 4 is given 
by s = b s: d m d 8 , where k2 = (b2 - a2) /b2 . It is because of this 
application that the integrals in this section are called elliptic integrals. 
Since there are many representations of elliptic functions, and many of 
them do not indicate the full functional dependence, symbolic and numerical 
tables of elliptic functions should be used carefully. 
References 
[l] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 
National Bureau of Standards, Washington, DC, 1964, Chapter 17, pages 
N. I. Akhiezer, Elements of the Theory of Elliptic Functions, Translations 
of Mathematical Monographs, Volume 79, Amer. Math. SOC., Providence, 
Rhode Island, 1990. 
[3] P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers 
und Physicists, Springer-Verlag, New York, 1954, pages 252-271. 
[4] B. C. Carlson, “Elliptic Integrals of the First Kind,” S I A M J. Math. Anal., 
8, No. 2, April 1977, pages 231-242. 
[5] B. C. Carlson, “A Table of Elliptic Integrals of the Second Kind,” Math. of 
Comp., 49, 1987, pages 595-606 (Supplement, ibid., S13-S17). 
[6] B. C. Carlson, “A Table of Elliptic Integrals of the Third Kind,’’ Math. of 
Comp., 51, 1988, pages 267-280 (Supplement, ibid., Sl-S5). 
[7] B. C. Carlson, “A Table of Elliptic Integrals: Cubic Cases,” Math. of Comp., 
53, No. 187, July 1989, pages 327-333. 
[8] S. Iyanaga and Y. Kawada, Encyclopedic Dictionary of Mathematics, MIT 
Press, Cambridge, MA, 1980. 
[9] D. F. Lauden, Elliptic Functions und Applications, Springer-Verlag, New 
York, 1989. 
[ l O ] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cam- 
bridge University Press, New York, 1962. 
587-6 26. 
[2] 
35. Frullanian Integrals 
Applicable to Integrals of a Special form. 
Yields 
An analytic expression for the integral. 
158 I11 Exact Analytical Met hods 
Idea 
A convergent integral can sometimes be written as the difference of 
two integrals that each diverge. If these two integrals diverge in the same 
way, then the difference may be evaluated by certain limiting processes. 
Procedure 
convergent integral 
A Special case will illustrate the general procedure. Consider the 
It is improper to write this integral as 
sin 32 00 3 sin x 00 
dx- a l -dx, 
X2 X2 
since both of these integrals diverge. It is proper, however, to write 
sin 32 r = ' l i m ( L O0 - d x - i 3sinx -dx), Ca 
4 6 4 0 X2 X2 
which can be written as (using y = 32 in the second integral) 
sin y Ca sin x 00 I = -1im 3 (L „ d x - l 6 T d y ) 4 640 
= - 4 6+o lim L3' - s:t dx 
= 3 lim L3' dx 
4 6+0 
= log3, 
sinx 1 
since - -+ - as x -0. 
Example 
x2 x 
The above procedure can be used to derive the general rule 
(35.1) 
(35.2) 
I" tanhax - tanh bx 
The integral J = dx is in the form of (35.2), 
X 
with f (x ) = tanhx. Si&e tanh(m) = 1 and tanh(0) = 0 we find that 
J = log(a/b). 
35. F’rullanian Integrals 159 
Notes 
[l] A modification of the formula in (35.2) is (See Ostrowski [3]) 
where M[f] = limt,, 1 J: f(z) dz and m[ f ] = limt-0 t Jt’ f(z) dz. 
A generalization of the formula in (35.2) is 
t 
[2] 
when U and v are positive absolutely continuous functions and the limits 
involving u/v are positive, and M[zf(z)] and m[zf(z)] exist. 
A different generalization of the formula in (35.2) is [3] 
This formula can be extended (See Hardy [2]) to 
JUm [f ( a 2 ) - f (ba:Q)] logN zda: 
n=O 
N + 1 
References 
[l] 
[2] 
[3] 
[4] 
E. B. Elliot, “On Some (General) Classes of Multiple Definite Integrals,” 
Proc. London Muth . SOC., 8, 1877, pages 35-47 and 146-158. 
G. H. Hardy, “A Generalization of Frullani’s Integral,” Messenger Mu th . , 
34, 1905, pages 11-18. 
A. Ostrowski, “On Some Generalizations of the Cauchy-F’rullani Integral,” 
Proc. Nutl. Acud. Sci. USA, 35, 1949, pages 612-616. 
W. Squire, Integrat ion f o r Engineers und Scientists, American Elsevier Pub- 
lishing Company, New York, 1970, pages 99-105. 
160 111 Exact Analyt ical Met hods 
36. Funct ional Equat ions 
Applicable to Definite integrals. 
Yields 
Sometimes an integral can be formulated as the Solution to a functional 
equation. (That is, an algebraic equation relating the unknown function at 
different values of the dependent variable.) 
Idea 
By manipulating an integral into a functional equation, it may be 
possible to evaluate the integral or to obtain information about the integral. 
Procedure 
There are many types of functional equations that may exist. There 
are no general rules on how best to proceed. 
Example 
Consider the integral 
I (u ) = iT log (1 + 2~ cos 2 + U') dx. (36.1) 
Breaking up the region of integration, Jt = 1;'' + J:", shows that I ( a ) = 
I ( -U) . Therefore: 
i= (36.2) 2I(a) = [log (I + 2acosz + U') +log (1 - ~ U C O S ~ + U ' ) ] dx 
= L'log (1 + a4 + 2a2(1 - 2 ~ 0 s ~ x)) dx. 
By performing many manipulations, such as 
log (1 + 2acosx + U') dx = 2 log (1 + 2a sin x + a') dx, 
the last integral in (36.2) can be shown to be equal to 
2I(a) = iT log (1 + 2a2 sin' x + a4) dx = I(a') . 
Hence, our functional equation is 2I(a) = I(.'). This is equivalent to 
1 
2" 
I ( a ) = -I (8). (36.3) 
Now note that I (0) = 0. When U' < 1, both terms on the right-hand side 
of (36.3) are approaching Zero as n -, 00. Therefore, we conclude that 
I ( a ) = o for U' < 1. 
36. Functional Equations 161 
Notes 
[l] The above example is from Squire [2 ] . 
evaluation of (36.1) is given by I ( u ) = 
In Book [l] the Problem was to Show that 
Squire Shows that the complete 
for u2 5 i 
{ O nlogu2 for u2 > i ' 
[2] 
d s 00 dt 
3/2 7r, where q ( t ) = i l log(1 + s t ) - 
1 + s2' 
The first Solution given in Book [l] started from the two functional equations 
for q( t ) : 
1 1 
q( t ) = q (?) + logt and q ( t ) + q(- t ) = log(1 - i t ) , 
where each functional equation has a different domain of applicability. 
Consider the integral I = [3] sin2 x dx . This integral can be manipulated 
into I = 
21 = JoT/2(cos2 x + sin2 x ) dx = 
cos2 x dx . Adding these two representations of I results in 
Ir 7r dx = - so that I = -. 
2 ' 4 
[4] Consider the integral 
00 
I = 1 s d x . 
Using the Change of variable z = l / x we obtain 
We conclude that I = 0. 
[5] Consider the integral 
dx . 
Using z = 7r/2 - x we find 
d z . 
Adding these two expressions for I results in 21 = 
I = 7r/4 (independent of n). 
d y = n / 2 . Hence, 
162 I11 Exact Analytical Methods 
Re ferences 
[l] 
[2] 
D. L. Book, Problem #6575, Amer. Math. Monthly, 97, No. 6, June-July 
1990, pages 537-540. 
W. Squire, Integration for Engineers und Scientists, American Elsevier Pub- 
lishing Company, New York, 1970, pages 85-87. 
37. Integration by Parts 
Applicable to 
Yields 
Single and multiple integrals. 
A reformulation of the integral. 
There is a simple integration by parts formula; it enables many inte- 
Idea 
grals to be evaluated exactly. 
Procedure 
The single integral I = s,” f(x) dx may often be Cast into the form 
I = J / ) u(x) dv(x). (37.1) 
If this is the case, then (37.1) may be evaluated by the integration by parts 
formula to obtain 
I = u(x)v(x)/ - 1 v(x)du(x). b b (37.2) 
a a 
Example 1 
If we have the integral 
J = L’xcosxdx 
we recognize that U = x and v = sinx (since du = coszdx). Hence, (37.2) 
may be used to obtain 
Y Y 
J = xsinx/o - Jo sinxdx (37.3) 
since du = dx. The last integral appearing in (37.3) is elementary and so 
(37.4) 
= ysiny +cosy - 1. 
37. Integration by Parts 163 
Example 2 
The Gamma function is defined by the integral r(z) = Jr tz- le- t dt. 
If we choose x = e-t and U = t Z / x (so that du = tZ-' dt) , then this integral 
has the form of (37.1). Hence, from (37.2) we have 
or r ( z + 1) = xr(z ) . Since we can easily determine that r(1) = 1, we 
conclude that, when n is a positive integer, r ( n + l ) = n.(n-l>. . . ..2.1 = n!. 
Hence, the Gamma functionis the generalization of the factorial function. 
Notes 
[l] The integration by parts formula may be re-applied to (37.2). For example, 
if f,, Stands for the n-th derivative of f and g 
of g then (See Brown [l]) 
b 
-k f2g3 
a 
, Stands for the n-th integral 
b 
+ .... (37.5) 
a 
[2] Green's theorem is essentially a multidimensional generalization of the usual 
integration by parts formula, since it relates the value of an integral to the 
values of some functions on the boundary of the region. See the section on 
line and surface integrals (page 164) for more details. 
Henrici [4] uses Cauchy's theorem to relate contour integrals to area inte- 
grals. For example, for the region R bounded by the curve r we have 
[3] 
f(x) dz = 22 11 afo dx dy, dz 
[4] Integration by parts may also be used to obtain an asymptotic expansion of 
an integral, See page 215. 
164 I11 Exact Analytical Methods 
References 
[l] 
[2] 
[3] 
[4] 
[SI 
[6] 
J . W. Brown, “An Extension of Integration by Parts,” Amer. Math. Monthly, 
67, No. 4, April 1960, page 372. 
P. S. Bullen, “A Survey of Integration by Parts for Perron Integrals,’’ J. 
Austral. Math. SOC. Ser. A , 40, No. 3, 1986, pages 343-363. 
U. Das, and A. G . Das, “Integration by Parts for Some General Integrals,” 
Bull. Austral. Math. SOC., 37, No. 1, 1988, pages 1-15. 
P. Henrici, Applied und Computational Complex Analysis, Volume 3, John 
Wiley & Sons, New York, 1986, pages 289-290. 
W. Kaplan, Advanced Calculus, Addison-Wesley Publishing Co., Reading, 
MA, 1952. 
G. B. Thomas, Jr., and R. L. Finney, Calculus und Analytic Geometry, 7th 
Edition, Addison-Wesley Publishing Co., Reading, MA, 1988. 
38. Line and Surface Integrals 
Applicable to 
Yields 
Line and surface integrals. 
A reformulation as an ordinary integral. 
Using a parameterization, line integrals and surface integrals can be 
Idea 
written as ordinary integrals. 
Procedure: Line Integrals 
A line integral is an integral whose path of integration is a path in n- 
dimensional space. For example, in two dimensions, if f(z, y) is continuous 
on the curve C, then the integrals sc f(z, y) dz and sc f(z, y) dy both exist. 
Here, C is either continuous, or Piece-wise continuous (in which case the 
above integrals are interpreted to be the sum of many integrals, each one 
of which has a smooth contour). 
Line integrals can be evaluated by reducing them to ordinary integrals. 
For example, if f(z, y) is continuous on C, and the integration contour is 
parameterized by (+(t),q(t)) as t varies from U to b, then 
In many applications, line integrals appear in the combination 
38. Line and Surface Integrals 165 
which is often abbreviated as 
(38.1) 
where the parentheses are implicit . 
If the vector U is defined by U = P ( x , y)i + Q(x , y)j, then the integral 
in (38.1) can be represented as J = sc UT ds, where ds is an element of 
arc-length, and UT = U - T denotes the tangential component of U (that 
is, the component of U in the direction of the unit tangent vector T, the 
sense given by increasing 3). Alternately, if the vector v is defined by 
v = Q ( z , y)i - P(z , y)j, then the integral in (38.1) can be represented as 
J = Jc vn ds, where vn = v - n denotes the normal component of v (that 
is, the component of v in the direction of the unit normal vector n which 
is 90” behind T). 
Path Independence 
I = 1 X dx + Y d y + 2 dz will be independent of the path in D 
Let X , Y , 2 be continuous in a domain D of space. The line integral 
0 if and only if U = ( X , Y, 2) is a gradient vector: U = grad F , where F 
is defined in D (that is, F, = X , Fy = Y , and F, = 2 throughout D ) , 
0 if and only if sc X dx + Y d y + 2 dz = 0 on every simple closed curve 
C in D. 
Green’s Theorems 
integral to an integral over an area (see Kaplan [l]): 
Green’s theorem, in its simplest form, relates a two-dimensional line 
Theorem: (Green’s) Let D be a domain of the zy plane and 
let C be a Piece-wise smooth simple closed curve in D whose 
interior R is also in D. Let Y(z,y) and Q(z,y) be functions 
defined in D and having continuous first partial derivatives 
in D. Then 
(38.2) 
Green’s theorem can be written in the two alternative forms (using 
U = P ( x , y)i + Q(z, y)j and v = Q(x , y)i - P ( x , y)j, as above): 
curl U dx d y JJ 
R 
$ v n d s = JJ divvdxdy. 
R 
( 3 8 . 3 4 
The second relation in (38.3) 
theorem is sometimes stated as 
is also known as Stokes’ theorem. This 
166 I11 Exact Analytical Methods 
Theorem: (Stokes) Let S be a piecewise smooth oriented 
surface in space, whose boundary C is a piecewise simple 
smooth simple closed curve, directed in accordance with the 
given orientation of S. Let U = Li+ Mj+Nk be a vector field, 
with continuous and differentiable components, in a domain 
D of space including S. Then, sc UT ds = ss (curlu - n) da, 
S 
where n is the Chosen unit normal vector on S. That is 
1 L dx + M d y + N d z = JJ(ay-=) aN d M d y d z 
S 
d L d N d M dL 
Green's theorem can be extended to multiply connected domains as 
follows: 
Theorem: Let P(x , Y ) and Q(x, Y ) be continuous and have 
continuous derivatives in a domain D of the plane. Let R 
be a closed region in D whose boundary consists of n dis- 
tinct simple closed curves {CI, Cz, . . . , Cn}, where CI includes 
{Cz, . . . ,Cn} in its interior. Then 
~6~ [Pdx + Q d y ] + f2 [Pda: + Q d y ] +. . . + [Pdx + Q ~ Y ] in 
dQ dP 
Specifically, if - = - in D, then dx d y 
Procedure: Surface Integrals 
normal vector n, then the surface integral 
If a surface S is given in the form z = f ( x , Y ) for ( x , Y ) in Rcy, with 
J J ( - ~ g - M & + N dxdy, J J L d y dz+M dz dx+N dx d y = z t af 1 
S R, aI 
with the + sign when n is the upper normal, and the - sign when n is the 
lower normal. If we define v = Li + Mj + N k then we may also write 
38. Line and Surface Integrals 
+ 
I 
I 
167 
- -> 
-1 I 1 
I 
Figure 38. The contour C for (38.4). 
where da is an element of surface area. Here, n = f(-f,i - f9j + 
k)/ J7 1 + f," + f2, with the + or - sign used according to whether n is 
the upper normal or lower normal. 
The generalization of (38.3.a) to 3 dimensions is known as the diver- 
gence t heorem, or as Gauss' theorem: 
Theorem: (Divergence) Let v = Li + Mj + Nk be a vector 
field in a domain D of space. Let L, M, and N be continuous 
and have continuous derivatives in D. Let S be a piecewise 
smooth surface in D that forms the complete boundary of a 
bounded closed region R in D. Let n be the outer normal of 
S with respect to R. Then 
J J v, d a = JJJ div v dx d y d z ; 
S R 
that is 
=/I/ (E d x + d y + 2) dz d x d y d z . 
R 
Example 1 
Consider t he integral 
I = (x3 - y3) dy (38.4) 
where C is the semicircle y = d w shown in Figure 38. The contour C 
can be represented parametrically by x = COS t and y = sin t for 0 5 t 5 T . 
Hence, the integral can be evaluated as 
3T I = JOT (cos3t - sin3t)costdt = - 
8 ' 
(38.5) 
168 I11 Exact Analytical Methods 
Alternatively, the integral could have been evaluated by using the x- 
parameterizat ion t hroughout 
I = l-’ (x3 - (1 - x 2 ) 3 / 2 ) ( -x ) dx. 
J3 
This integral, which looks more awkward, is equivalent to (38.5) under the 
Substitution x = cost. 
Example 2 
Here are a few examples of Green’s theorem: 
Consider the integral K = jC [ (y2 + sinx2) dx + (COS y2 - x) dy] , where 
C is the boundary of the unit Square ( R := (0 5 x 5 1, 0 5 y 5 1)). A 
direct evaluation of this integral by parameterizing C is quite difficult. 
For example, using z as the Parameter on the bottom piece of C, 
{y = 0, 0 5 x 5 1}, necessitates the evaluation of the integral Jt sin x 2 dx, which is not elementary. However, using Green’s theorem 
in (38.2) (with P = y2 + sinx2 and Q = COS y2 - x) we may write this 
integral as K = SJ (-1 - 23) dx dy. As a Set of iterated integrals, we 
readily find that K = -2. 
Let C be the circle x2 + y2 = 1. Then, using(38.2) 
R 
f [4xy3 dz + 6x2y2 dy] = (12xy2 - 12zy2) dz dy = 0. JJ R 
Let C be the ellipse x2 + 4y2 = 4. Then, using (38.2) 
= 2(area of ellipse) = 47r. 
Notes 
When the contour of integration in a line integral is closed, then we often 
represent the integral by the Symbol 9, rather than the usual J. 
When the contour in a two-dimensional line integral is closed, and the 
integrand is analytic, then contour integration techniques may be used (See 
page 129). Sometimes a two-dimensional line integral can be extended to a 
closed contour and be evaluated in this manner. (Note, however, that the 
integrand in (38.4) is not analytic.) 
The evaluation of the integral in Example 2.C required that an area be 
known. Using Green’s theorems, we can write the following integral expres- 
sions for the area bounded by the contour C: 
38. Line and Surface Integrals 169 
[4] If D is a three-dimensional domain with boundary B, let dV represent the 
volume element of D , let ds represent the surface element of B, and let 
dS = ndS, where n is the outer normal vector of the surface B. Then 
Gauss’s formulas are (See Iyanaga and Kawada [3], Page 1400) 
///VAdV = // dS . A = // ( n . A ) d S 
D B B 
///V x AdV = // dS x A = // (n x A ) d S 
D B B 
D B 
where q5 is an arbitrary scalar and A is an arbitrary vector. 
There are also Green’s theorems that relate a volume integral to a surface 
integral. Let V be a volume with surface S, which we assume to be simple 
and closed. Define n to be the outward normal to S. Let q5 and + be scalar 
functions which, together with V2q5 and V2+, are defined in V and on S. 
Then (See Gradshteyn and Ryzhik [2], Page 1089) 
(A) Green’s first theorem states that 
[5] 
#g dS = s, (q5V2+ + Vq5 - V+) dV. 
(B) Green’s second theorem states that 
Re ferences 
[ 11 
[2] 
131 
[4] 
W. Kaplan, Adwanced Calculus, Addison-Wesley Publishing Co., Reading, 
MA, 1952. 
I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, 
Academic Press, New York, 1980. 
S. Iyanaga and Y. Kawada, Encyclopedic Dictionary of Mathematics, MIT 
Press, Cambridge, MA, 1980. 
G. B. Thomas, Jr., and R. L. Finney, Calculus and Analytic Geornetry, 7th 
Edition, Addison-Wesley Publishing Co., Reading, MA, 1988. 
170 I11 Exact Analytical Methods 
39. Look Up Technique 
Applicable to Integrals of certain forms. 
Yields 
nique. 
An exact evaluation, an approximate evaluation, or a numerical tech- 
Idea 
Many integrals have been named and well studied. If a given integral 
can be transformed to a known form, then information about the evaluation 
may be obtained from the appropriate reference. 
Procedure 
Compare the integral of interest with the lists on the following pages. 
If the integral of interest appears, See the reference cited for that integral. 
There are four lists of integrals, those with no Parameters (i.e., con- 
stants), those with one Parameter, those with two Parameters, and those 
with three or more Parameters. 
Notes 
The integrals in this section cannot be evaluated, in closed form, in terms 
of elementary functions (see page 77). 
Some of the integrals are only defined for some values of the Parameters; 
these restrictions are not listed in the tables. For example, the gamma 
function r ( x ) is not defined when x is a negative integer. 
Realize that the Same integral may look different when written in different 
variables. A transformation of your integral may be required to make it look 
like one of the forms listed. 
In this section, the references follow the listings of integrals. 
If the integral desired is not of a common form, then it will not appear in 
this section. However, it might be tabulated in one of the tables of integrals, 
See Page 190. 
39. Look Up Technique 171 
Constants Defined by Integrals 
Catalan’s constant (See Lewin [21], Page 34) 
00 
tan-l t 
G = Ti&) = J/’t dt = (-’)’ - 0.91596559. 
k=O (2k+ 1)2 - 
Euler’s constant (See Gradshteyn and Ryzhik [15], 8.367.4, Page 946) 
7 = - lm eAt logt dt z 0.577215. 
Unnamed - related to random Permutations (See Goh and Schmutz [14]) 
2 = lrn log log (s) dt 2 1.11786. 
Integrals with One Parameter 
Airy function (See Spanier and Oldham [26], 56:3:1, Page 555) 
00 
Ai(z) = ‘1 COS ( St3 + zt ) d t . 
Ir 
Related to Airy’s function (See Abramowitz and Stegun [l], 10.4.42, Page 448) 
Gi(z) = ilOO sin ( i t 3 + z t ) d t . 
Related to Airy’s function (See Abranrowitz and Stegun [l], 10.4.44, page 448) 
Hi(z) = L l m Ir exp ( f t 3 + z t ) d t . 
Bairy function (See Abramowitz and Stegun [1], 10.4) 
Bi(z) = Ir cos (f + z t ) exp (-: + x t ) d t . 
Binet integrals (See Van Der Laan and Temme [19], page 122) 
Bloch-Gruneisen integral (See Deutsch [9]) 
d t . 
t5 z ( x ) = 
172 I11 Exact Analytical Met hods 
Clausen's integral (See Lewin [21 , Chapter 4, pages 91-105) 
dt . 
Cosine integral (See Abramowitz and Stegun [l], 5.2.2, page 231) 
cost - ldt* Ci(z) = y + log z + 
Auxiliary cosine integral (See Spanier and Oldham [26], 38:13:2, page 371) 
gi(z) = 1,- dt. 
t2 + 1 
Hyperbolic cosine integral (See Abramowitz and Stegun [l], 5.2.4, page 231) 
Chi(z) = y + logz + LZcosh: - dt . 
Dawson's integral (See Spanier and Oldham [26], Chapter 42, pages 405-410) 
00 
daw(z) = 1 e t2-x2 dt. 
Dilogarithm (See Lewin [21]) 
Li2(z) = - lx:); - dt. 
Error function (See Abramowitz and Stegun [l], 7.1.1, page 297) 
erf(z) = % L z e - t 2 dt. 
Complementary error function (See Abramowitz and Stegun [l], 7.1.2, page 297) 
2 , erfc(z) = /nJi e-t2 dt. 
Complete elliptic integral of the first kind (See page 154) 
Complete elliptic integral of the second kind (See page 154) 
~ ( k ) = JT12 J-dO. 
0 
Exponential integral (See Abramowitz and Stegun [l], 5.1.1, page 228) 
00 -t 
Ei ( z ) = 1 y d t . 
39. Look Up Technique 173 
Exponential integral (See Abramowitz and Stegun [l], 5.1.2, page 228) 
Ei(a) = -fr$dt. 
Fresnel integral (See Abramowitz and Stegun [l], 7.3.1, Page 300) 
C(x) = 1’ COS ( a t 2 ) dt. 
Fresnel integral (See Abramowitz and Stegun [l], 7.3.2, page 300) 
S(x) = 1’ sin ( q t 2 ) dt. 
Gamma function (See Abramowitz and Stegun [i], 6.1.1, page 255) 
r ( z ) = lwt’- le- t dt. 
Product of Gamma functions (See Abramowitz and Stegun [l], 6.1.17, page 256) 
Inverse tangent integral (See Lewin [21], Chapter 2, pages 33-60) 
Ti2(x) = ix- dt. 
Lebesgue constants (See Wong [27], page 40) 
Legendre’s Chi function (See Lewin [21], page 17) 
Logarithmic integral (See Abramowitz and Stegun [l], 5.1.3, page 228) 
Psi (Digamma) function (See Abramowitz and Stegun [l], 6.3.21, Page 259) 
dt . 
Phi function (normal probability function) (See Abramowitz and Stegun [l], 
26.2.2, Page 931) 
@(a) = - e-t2/2 dt. 
174 I11 Exact Analyticai Methods 
Riemann's Zeta function (See Abramowitz and Stegun [l], 27.1.3, page 998) 
Sine integral (See Abramowitz and Stegun [l], 5.2.1, Page 231) 
Si(z) = L Z T d t . sin t 
Auxiliary sine integral (See Spanier and Oldham [26], 38:13:1, page 370) 
Hyperbolic sine integral (See Abramowitz and Stegun [l], 5.2.3, Page 231) 
Shi(z) = i z t d t . sinh t 
Trilogarithm (See Lewin 1211, Chapter 6, pages 136-168) 
Lis(z) = J/'Y dt. 
Integrals 
a-Order Green's 
1 
Ga(x) = 6 
with Two Parameters 
function (See Iyanaga and Kawada [16], page 165B) 
Anger function (See Abramowitz and Stegun [l], 12.3.1, page 498) 
J,(z) = :LTcos(u8 - zsin8)dB. 
Bessel function (See Abramowitz and Stegun [ l ] , 9.1.22, Page 360) 
dt . sin u r -2sinht-ut J&) = cos(z sin8 - W O ) d8 - - 
Bessel function (See Abramowitz and Stegun [l], 9.1.22, page 360) 
Yv(z) = i;l" sin(zsin8-uO) {eut + e-vt COS(UT)} e-zsinht-vt dt. 
Beta function (See Abramowitz and Stegun [l], 6.2.1, page 258) 
B(z, w) = J l l t Z - 1 ( 1 - t y - l dt . 
39. Look Up Technique 175 
Bickley function (see Amos [2]) 
s," Kin-l(t) dt for n = 1 ,2 , . . . 
for n = 0 Kin(x) = { Ko(x) 
Debye function (see Abramowitz and Stegun [l], 27.1.1,Page 998) 
et - i 
Dnestrovskii function of index q (see Robinson [25]) 
Elliptic integral of the first kind (See Page 
dt 
J(1 - t2) ( l - k 2 t 2 ) 
sin 
147) 
Elliptic integral of the second kind (see Page 147) 
E(4 , k ) = lsin ' ./=~ 1 - k2 t2 dt. 
1 - t 2 
Repeated integrals of the error function (see Abramowitz and Stegun [l], 7.2.3, 
page 299) 
i"erfc(z) = inl 2 - ( t e z ) n t 2 d t . 
n! 
Exponential integral (See Abramowitz and Stegun [l], 5.1.4, Page 228) - e - z t 
&(z ) = J1 T d t . 
Generalized exponential integral (see Chiccoli et al. [7]) 
O0 e-zt E&) = J/ T d t . 
Fermi-Dirac integral (see Fullerton and Rinker [ l l ] ) 
F,(a) = J" 2 d t . 
i + e t - " 
Logarithmic Fermi-Dirac integral (See Fullerton and Rinker [ 111) 
Generalized F'resnel integral (See Abramowitz and Stegun [l], 6.5.7, page 262) 
C(a, U) = l*;t.-' cos t d t . 
176 I11 Exact Analytical Met hods 
Generalized F'resnel integral (See Abramowitz and Stegun [l], 6.5.8, page 262) 
S(Z, a ) = l m t a - ' sint dt. 
Hubbell rectangular-source integral (See Gabutti et al. [12]) 
Incomplete Gamma function (See Abramowitz and Stegun [l], 6.5.3, page 260) 
r ( a , Z) = l m t a - ' e - t dt. 
Generalized inverse tangent integral (See Lewin [21], Chapter 3, pages 61-90) 
Ti2(z, U) = L'- dt. 
Hurwitz function (See Spanier and Oldham 1261, 64:3:1, page 655) 
m U-i -ut t e dt . C(u;u) = - 
Repeated integrals of KO (See Abramowitz and Stegun [l], 11.2.10, page 483) 
00 e - ~ cosh t 
Ki,(z) = - dt . 
Kummer's function (See Lewin 
coshr t 
Legendre function of the first kind (See Spanier and Oldham [26], 59:3:1, page 
583) 
E(z) = 'Jla 7r [Z + ~Gcos~]~ dt. 
Legendre function of the second kind (See Spanier and Oldham [26], 59:3:2, page 
583) 
QU@) = IM [Z + J = C O S ~ ~ ] - ~ - ' dt. 
0 
Pearcey integral (See Kaminski [18]) 
t2 M 
P(z, y) = 1 exp{ i (\ + z y + Yt> } dt. 
-00 
Polygamma function (See Abramowitz and Stegun [l], 6.4, page 260) 
39. Look Up Technique 177 
Log-sine integral of Order n (See Lewin [21], page 243) 
e 
Ls,(8) = -1 log"-'12sin:l d8. 
Extended log-sine integral of the third Order of argument 8 and Parameter (Y (See 
Lewin [21], page 243) 
Lss(8,cr) = -le log 12sin 51 log 12sin (5 8 a + z) I d8. 
Sievert integral (see Abramowitz and Stegun [l], 27.4, page 1000) 
Polylogarithm of Order n (see Lewin [21], page 169) 
Struve function (See Abramowitz and Stegun [l], 12.1.7, page 496) 
sin(z COS 8) sin2" 8 d8. J;;r (Y+ $) H u ( z ) = 
Modified Struve function (see Abramowitz and Stegun [l], 12.2.2, page 498) 
sinh(z COS 8) sin2" 8 d8. J;;r (Y+ +) L ( z ) = 
Voigt function (see Lether and Wenston [20]) 
Weber function (see Abramowitz and Stegun [l], 12.3.3, page 498) 
E,(z) = :lT sin(v8 - z sin 0) d8. 
Unnamed integral (See Bonham [5]) 
f in (z ) = (-1)" 1' &(t) sinztdt. 
Unnamed integral (See Bonham [5]) 
fin+1(z> = (-1)"+' 1' &+l( t ) co~z td t . 
178 I11 Exact Analytical Methods 
Unnamed integral (See Chahine and Narasimha [6]) 
Unnamed integral (See Glasser [13]) 
Iu(x) = 1' t"(1 - t)"l sinxtl dt. 
Unnamed integral (See Nagarja [23]) 
Integrals with Three or More Parameters 
Incomplete Beta function (See Abramowitz and Stegun [l], 6.6.1, page 263) 
B,(a,b) = l"t"-'(l - t)b-' dt. 
Elliptic integral of the third kind (See page 147) 
dt 
(1 + nt2) J(1 - t"(1 - k2t2) - 
Double Fermi-Dirac integral (See Fullerton and Rinker [ l l ] ) 
SU 
00 tu 00 
Generalized Fermi-Dirac integral (See Pichon [24]) 
Gauss function (See Spanier and Oldham [26], 60:3:1, page 600) 
tb-' dt F ( a , b, c; X) = 
Incomplete hyperelliptic integral (See Loiseau, Codaccioni, and Caboz [22]) 
dx lX J a - X2x2 - Xnxn H ( a ) X ; X2, An) = 
180 I11 Exact Analytical Methods 
Unnamed integral (See Cole [SI) 
exp[-s(t + ycost - zsint)] dt. 
Unnamed integral (See Kölbig [17]) 
z(v, X, m) = z”-’(l- z)-’ log” z dz. 1‘ 
References 
[l] 
[2] 
M. Abramowitz and I. A. Stegun, Hundbook of Muthemuticul Functions, 
National Bureau of Standards, Washington, DC, 1964. 
D. E. Amos, “Algorithm 609: A Portable FORTRAN Subroutine for the 
Bickley Functions Kin(z),” ACM Truns. Muth. Software, 9, No. 4, December 
1983, pages 480-493. 
K. Aomoto, “On the Complex Selberg Integral,’’ Quart. J. Muth. Oxford, 
38, No. 2, 1987, pages 385-399. 
[3] 
. P o 0 1 duun-2 D. G. Anderson and H. K. Macomber, “Evaluation of - 1 [4] 
& o 
1 
exp{ -2(u - p)2 - g},,, J. Muth. und Physics, 45, 1966, pages 109-120. 
R. A. Bonham, “On Some Properties of the Integrals si P2n(t) sinzt dt and 
P2n+1 (t) COS z t dt,” J. Muth. und Physics, 45, 1966, pages 331-334. 
M. T. Chahine and R. Narasimha, “The Integral Jom un exp [ - ( U - u ) ~ - , / U ] 
du,” J. Muth. und Physics, 43, 1964, pages 163-168. 
C. Chiccoli, S. Lorenzutta, and G. Maino, “Recent Results for Generalized 
Exponential Integrals,’’ Comp. 43 Muths. with Appls., 19, No. 5, 1990, pages 
[8] R. J. Cole, “Two Series Representations of the Integral som exd-s(++y cos + 
- z sin +)] d+,” J. Comput. Physics, 44, 1981, pages 388-396. 
[9] M. Deutsch, “An Accurate Analytic Representation for the Bloch-Gruneisen 
Integral,’’ J. Phys. A: Muth. Gen., 20, 1987, pages L811-L813. 
[10] H. E. Fettis, “On the Calculation of Wu’s Integral,” J. Comput. Physics, 
53, 1984, pages 197-204. 
[ll] I. W. Fullerton and G. A. Rinker, “Generalized Fermi-Direc Integrals - FD, 
FDG, FDH,” Comput. Physics Comm., 39, 1986, pages 181-185. 
[12] B. Gabutti, S. L. Kalla, and J. H. Hubbell, “Some Expressions Related to the 
Hubbell Rectangular-Source Integral,” J. Comput. Appl. Muth., 37, 1991, 
pages 273-285. 
s”(1 - S ) ~ I sinzsl ds with an 
Application to Schlomilch Series,” J. Muth. und Physics, 43, 1964, pages 
[14] W. M. Y. Goh and E. Schmutz, “The Expected Order of a Random Permu- 
[15] I. S. Gradshteyn and I. M. Ryzhik, Tubles of Integrals, Series, und Products, 
[16] S. Iyanaga and Y. Kawada, Encyclopedic Dictionury of Muthemutics, MIT 
U 
[5] 
[6] 
[7] 
21-29. 
[13] M. L. Glasser, “A Note on the Integral 
158-162. 
tation,” Bull. London Muth. SOC., 23, 1991, pages 34-42. 
Academic Press, New York, 1980. 
Press, Cambridge, MA, 1980. 
Special Integration Techniques 181 
K. S. Kölbig, “On the Integral s: x”-’(l - x)-’ log” x dx,” J. Comput. 
Appl. Math., 18, No. 3, 1987, pages 369-394. 
D. Kaminski, “Asymptotic Expansion of the Pearcey Integral Near the 
Caustic,” SIAM J. Math. Anal., 20, No. 4, July 1989, pages 987-1005. 
C. G. van der Laan and N. M. Temme, Calculation of Special Functions: 
The Gamma function, the Exponential Integrals und Error-like Functions, 
Centrum voor Wiskunde en Informatica, Amsterdam, 1984. 
F. G. Lether and P. R. Wenston, “The Numerical Computation of the 
Voigt Function by the Corrected Midpoint Quadrature Rule for (-00, CO),” 
J. Comput. Appl. Math., 34, 1991, pages 75-92. 
L. Lewin, Dilogarithms und Associated Functions, MacDonald & Co., Lon- 
don, 1958. 
J. F. Loiseau, J. P. Codaccioni, and R. Caboz, “Incomplete Hyperelliptic 
Integrals and Hypergeometric Series,” Math. of Comp., 53, No. 187, July 
1989, pages 335-342. 
K. S. Nagarja, “Concerning the Value of 
ics, 44, 1965, pages 182-188. 
B. Pichon, “Numerical Calculation of the Generalized Fermi-Direc Inte- 
grals,” Comput. Physics Comm., 55, 1989, pages 127-136. 
P. A. Robinson, “Relativistic Plasma Dispersion Functions: Series, Integrals, 
and Approximations,” J. Math. Physics, 28, No. 5, 1987, pages 1203-1205. 
J. Spanier and K. B. Oldham, A n Atlas of Functions, Hemisphere Publishing 
corporation, New York, 1987. 
R. Wong, Asymptotic Approximation of Integrals, Academic Press, New 
York, 1989. 
P e-u2 
- du,” J. Math. und Phys- 1 u + x 
40. Special Integration Techniques 
Applicable to 
of which works on a Special class of integrals. 
There are many specialized integration techniques, each 
Integrands involving solut ions of second Order differential equat ions 
Consider the integral 
(40.1) 
where y1and y2 are linearly independent solutions to the ordinary differ- 
ential equation: 9‘‘ = Q(z)y. (Note that the Wronskian of y1 and y2 is a 
constant; W(yl,y2) = c # 0.) For this method to work, we require that f 
be homogeneous of degree -2, that is f(ay1, ay2) = ~ - ~ f ( y l , y2). In this 
182 I11 Exact Analyt ical Met hods 
case we find 
(40.2) 
where U = y2/yi. 
Example 1 
The following example is from Ashbaugh [3]. The integral 
sin2 x 
C O S ~ x + COS x sin3 z dx J = J 
can be evaluated by identiSling y1 = COS x and y2 = sin x as being solutions 
to y" + y = 0. Hence, we write J as 
= ; i o g l l + (:)31+~ 
=$log 1 1+ ( s i n x ~ ~ / - +c COS x 
= $ l o g I i + t a n 3 x l + ~ 
where C is an arbitrary constant. 
Example 2 
The following example is from Ashbaugh [3]. The integral 
dx Ca 
= 1 (Ai(x) - iBi(x))2 
(where Ai (Bi) is the Airy (Bairy) function) has the form of (40.1), with 
yi(x) = Ai(x) - i Bi(x) and y2(x) = Bi(x). We can write K as (recall that 
40. Special Integration Techniques 183 
Ai and Bi satisfy Airy's equation: y" = xy) 
YlY; - d Y 2 * 00 dx 00 
= 1 (Ai(x) - iBi(x))2 = 1 7r-' Y? 
= n - ( 
Integration of Rational Functions 
A rational function can always be integrated into a sum of rational 
functions and logarithmic (or arctangent) functions. Using D ( X ) and N(x) 
to represent polynomials, a rational function can be written using partial 
fractions as 
where the { r i } are the roots of D(x) = 0 (assumed here to be distinct). 
This integrand can be directly integrated to obtain 
dx "ri) J (x - r i ) n j Z i ( r i - f(x) dx = (polynomial in x) dx + 
N ( T ~ ) log(x - ri) 
J J 
= (different polynomiai in x) + 
i nj&i - T j ) * 
(40.3) 
This is formally correct, even when the roots are complex. Since logarithms 
and arctangents are related by log(u - iv) = 22 tan-l(u/v>, the expression 
in (40.3) is always applicable. 
Example 3 
dx J & = J (x - i ) (x + i ) (x - i>(x + i) 
= + log(x - 1) + + log(x + 1) + E log(x - i) + & log(x + i) 
1 2-1 1 
4 x + l 2 
= -log - - - tan-l x. 
For the case when some of the roots are repeated, see Stroud [6]. 
184 I11 Exact Analyt ical Met hods 
Use of Infinite Series 
Sometime an integral may be evaluated by expanding the integrand in 
a series, integrating term by term, and then re-summing the result. This 
technique may also be used to obtain an asymptotic expansion, see Wong [7] 
for details. 
Example 4 
sin ax 
Consider the integral I = dx. Formally expanding the 
denominator, and interchanging i&egration and Summation results in 
I = 5 lm e-nx sinax dx 
n=l 
00 a - -E- 
= E 2 (cotaa - $) 
n=l 
where we used Jolley [4] to recognize the cotangent sum. 
Example 5 
Consider the integral 
I = lm e-az2Jl(ßz) dx (40.4) 
where J1 is a Bessel function. Rom Abramowitz and Stegun [l] 9.1.10, we 
note t hat 
(40.5) 
Using (40.5) in (40.4), and interchanging Orders of integration (see page log), 
results in 
ß " 1 
= 5 E k ! ( k + l ) ! 
k=O 
k + l 1 " 
(k + l)! 
where we had to recognize the series for the exponential function. 
40. Special Integration Tec.hniques 185 
Series Transpositions 
re-summing. An example will demonstrate the general ideas. 
Squire [5] presents an elegant technique using series expansions and 
sin x 
Consider the integral I = dx. The infinite integration region 
can be written as an infinite sÜm of finite integration regions, and then 
each finite region of integration can have the variables changed: 
k7r 
sin U COS - sin U sin - ' ) d u 
2 + 
k=O i k n + u i ( k + l ) n - u 
sin - 
k=O 
1 + . . . I + "- - 1 +..-I> =l T I 2 d u s i n u x ( [ ' - - - 00 
U T + U I r - U 2n-U 
k=O 
du sinu (L) = 7r 
du = - 
sin U 2 
where the changes of variables used were U = x-k7r/2 and then U = 7r/2-u. 
The final Summation was found in Jolley [4], Summation #769. 
Note 
[l] Arora et al. [2] notice that the two properties 
(A) soa f(x) dx = soa/2 f(x) dx + s;I2 f (a - X) dx 
(BI soa f ( 4 dx = soa f (a - 2) dx, 
if used cleverly, can evaluate some sophisticated integrals. They evaluate 
Ir dx = -- to demonstrate this. 4 
186 
Re ferences 
I11 Exact Analytical Methods 
111 
PI 
[31 
141 
151 
171 
161 
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 
National Bureau of Standards, Washington, DC, 1964. 
A. K. Arora, S. K Goel, and D. M. Rodriguez, “Special Integration Tech- 
niques for Trigonometric Integrals,” Amer. Math. Monthly, February 1988, 
Vo195, No. 2, pages 126-130. 
M. S. Ashbaugh, “On Integrals of Combination of Solutions of Second-Order 
Differential Equations,” J. Phys. A : Math. Gen., 19, 1986, pages 3701- 
3703. 
L. B. W. Jolley, Summation of Series, Second Edition, Dover Publications, 
Inc., New York, 1961. 
W. Squire, Integration for Engineers und Scientists, American Elsevier Pub- 
lishing Company, New York, 1970, Page 92. 
A. H. Stroud, Approximate Calculation of Multiple Integrals, Prentice-Hall 
Inc., Englewood Cliffs, NJ , 1971, Section 2.4, pages 49-52. 
R. Wong, Asymptotic Approximation of Integrals, Academic Press, New 
York, 1989, Chapter 4, pages 195-240. 
41. Stochastic Integration 
Applicable to 
Yields 
Idea 
are Chosen in a limiting process. 
Background 
Suppose that w ( t ) is the Wiener process and G = G ( t ) = G ( t , w ( t ) ) 
is an arbitrary function of w(t) and the time t. (See the Notes section for 
information about the Wiener process.) The stochastic integral 
Integrals in which the measure involves Brownian noise. 
Information on how to evaluate Ito and Stratonovich integrals. 
Different types of stochastic integrals exist, depending on how Points 
(41.1) 
is defined as a kind of Riemann-Stieltjes integral. That is, first divide the 
interval [to,t] into n sub-intervals: t o 5 tl 5 t 2 5 . - . 5 tn-1 5 tn = t. 
Then choose the Points {ri}, for i = 1 , 2 , . . . , n, such that ri lies in the 6 th 
sub-interval: ti-1 5 ri 5 ti. The stochastic integral I is now defined as a 
limit of partial Sums, I = limn4, Sn , with 
41. Stochastic Integration 187 
Note that this limit depends On the particular choice of the intermediate 
Consider, for example, the Special case of G(t) = w(t ) . Then we find 
Points {Ti}. 
the expectation of Sn to be 
E 
i= 1 
If, for example, we take Ti = ati + (1 - a)t+l, where 0 < a < 1, then 
E [ Sn] becomes 
(41.3) 
Clearly, the value of Sn (and hence I ) , in this example, depends on a. 
For consistency, some specific choice must be made for the Points {Ti}. 
For the Ito stochastic integral we choose Ti = ti-1 (i.e., a = 0 in the above). 
We show this by use of the notation $. That is 
(41.4) 
where ms-lim refers to the mean square limit. 
188 I11 Exact Analytical Methods 
Example 
Suppose we would like to evaluate the Ito stochastic integral 
(41.5) 
If we write wi for w(ti) then (41.4) becomes (using G(s) = w(s)) 
6: W ( S ) ~ w ( s ) = rns-limn+, Sn with 
n 
i= l 
n 
i= 1 
n 
1 
2 [(Wi-l + A w ~ ) ~ - ( ~ i - 1 ) ~ - ( A w ~ ) ~ ] 
i= 1 
n n 
f [ ( ~ i ) ~ - ( w i - ~ ) ~ ] - i C ( A W ~ ) ~ 
i= 1 i= 1 
n 
f [w2(t) - w2(to)] - f C ( A W ~ ) ~ 
i= l 
where Awi = wi - wi-1. Now the mean-Square limit of f Cy'l(Awi)2 can 
be shown to be i ( t - t o ) . Hence, 
= f [W2(t) - W2(tO) - (t - to)] . 
Note that the result is not the same result that we would have obtained 
by the usual Riemann-Stieltjes integral (in which the last term would be 
absent). Note that the expectation of (41.6) yields the value 0, which is 
the same value as given by (41.3). 
41. Stochastic Integration 
Notes 
189 
The Wiener process is a Gaussian random process that has a fixed mean 
given by its starting Point, E[w(t)] = wo = w(to), and a variance of 
E[(w( t ) - = t - t o . From this we can compute that E[w(t)w(s)] = 
min(t, s). The sample paths of w(t) are continuous, but not differentiable. 
We define the Stratonovich stochastic integral (indicated by use of the 
notation $) to be (SeeSchuss [2]) 
It can be shown that the 
, t i-1 [W(t i ) - w(ti-l)] * ) 1 
Stratonovich integral has the usual properties 
of integrals. In particular, we have the fundamental theorem of integral 
calculus, iot f’(4s)) d W = f ( w W - f(w(toN7 
integration by Parts, etc. Taking the Stratonovich integral of the integrand 
in (41.5) results in 6; w(s) dw(s) = f [w2( t ) - w2( to ) ] . 
For arbitrary functions G, there is no connection between the Ito integral 
and the Stratonovich integral. However, when z( t ) satisfies the stochastic 
differential equation dz( t ) = a[z ( t ) , t] dt + b[z ( t ) , t] dw(t) , it can be shown 
that (See Gardiner [l]) 
This relates, in a way, the Stratonovich integral and the Ito integral. 
Stochastic integration can also refer to the (ordinary) integration of ran- 
dom variables. Since the linear Operations of integration and expectation 
commute, the following results are straightforward to derive. Let { X(t)} 
be a continuous Parameter stochastic process with finite second moments, 
whose mean (m(t) = E[X(t)]) and covariance ( K ( s , t ) = Cov[X(s),X(t)]) 
are continuous functions of s and t . Then 
E [ [ X(t) d t ] = Lb m(t ) dt 
E [ I[X(t)dtl2] = JlbLbE[X(2?)X(t)]dtdS 
Var [ [ X(t) d t ] = [ [ K(s , t ) dt ds. 
42. Tables of Integrals 191 
0 Dwight [5] has a recursion for the indefinite integral in number 
480.9: 
- J tann-2 x dx. tann-l x tann x dx = J n - 1 
0 In Gradshteyn and Ryzhik [7], the definite integral appears as 
number 3.622.1 : 
PT [when] IRep) < 1. tg*p x dx = - sec - 7r Jlr/z 2 2 
0 In Gradshteyn and Ryzhik [7], the indefinite integral appears as 
numbers 2.527.2, 2.527.3, and 2.527.4: 
tgp-lx 
P - 1 
J tgpx dx = - - J tgp-2 x dx [when] 
-(-1 
n (-1)k-1 tg2n-2k+2 
2n - 2k + 2 tgan+l xdx = 
k = l 
P Z 1 
In COS x 
0 In Gröbner and Hofreiter [8] the definite integral appears as num- 
ber 331.31: 
7r 
[when] -1 < X < 1. X7r 
2 COS - 
0 In Prudnikov, Brychov, and Marichev [12], the integral appears 
tg' xdx = Jlrlz 2 
as number 2.5.26.7: 
7r 
[when] I Repl < 1. LTl2 tgp x dx = 2 cos(p7r/2) 
0 In Prudnikov, Brychov, and Marichev [12], the indefinite integral 
appears as numbers 1.5.8.1, 1.5.8.2, and 1.5.8.3: 
/(tgx)Pdx = & T ( t g x ) P - l 1 - J(tg xlp-2 dx 
P - 
192 
Notes 
I11 Exact Analytical Met hods 
Reaiize that the Same integral may look different when written in different 
variables. A transformation of your integral may be required to make it look 
like one of the form listed. 
It is not always clear that having a symbolic evaluation of an integral is 
more useful than the original integral. For example, it is straightforward to 
Show that 
Given a specific value of x, numerically determining I(x) by using this 
formula requires the computation of logarithms and inverse tangents. In 
some cases it might be easier to approximate I(x) numerically directly from 
its definition. 
It is an unfortunate fact that a not insignificant fraction of the tabulated 
integrals are in error. See, for example, Klerer and Grossman [9]. 
A common error is to produce a discontinuous antiderivative when 
a continuous integral is available. For example, the symbolic Computer 
language REDUCE produces (See page 117): 
I = J’A 
2 + cosx 
A s i n x 
- - -- 2fi arctan (x) + arctan ( ) + 5 3 cosx + 1 ~ ( C O S Z + 1) & * 
This (correct) antiderivative is continuous, yet Abramowitz and Stegun [l], 
4.3.133, report the discontinuous result - 2 I = -arctan- tan(x/2). These 
& d3 
antiderivatives agree on the interval -?r < x < T , but 7 is periodic while I 
is not. 
If an integral is recognized to be of a certain form, then appropriate tables 
may be used. For example, an integral of the form I = s-”, f(x)e-xt dx 
represents a Fourier transform of the function f(x). Hence, a table of Fourier 
transforms (such as Oberhettinger [ll]) might be an appropriate place to 
look for an evaluation of I. 
Note that Oberhettinger [ll] has tables of Fourier transforms, Fourier sine 
transforms, and Fourier cosine transforms. 
All of the integral evaluations in Gradshteyn and Ryzhik [7] are referenced, 
so that one level of checking against typographic errors can be performed. 
42. Tables of Integrals 
References 
193 
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 
National Bureau of Standards, Washington, DC, 1964. 
A. Apelblat, Table of Definite und Indefinite Integrals, American Elsevier 
Publishing Company, New York, 1983. 
W. H. Beyer (ed.), CRC Standard Mathematical Tables und Formulae, 29th 
Edition, CRC Press, Boca Raton, Florida, 1991. 
G. P. Bois, Tables of Indefinite Integrals, Dover Publications, Inc., New York, 
1961. 
H. B. Dwight, Tables of Integrals und Other Mathematical Data, The MacMil- 
lan Company, New York, 1957. 
Staff of the Bateman Manuscript Project, A. Erdelyi (ed.), Tables of Integral 
‘Pransforms, in 3 volumes, McGraw-Hill Book Company, New York, 1954. 
I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, und Products, 
Academic Press, New York, 1980. 
W. Gröbner and N. Hofreiter, Integralyafel, Springer-Verlag, New York, 
1949. 
M. Klerer and F. Grossman, “Error Rates in Tables of Indefinite Integrals,” 
Indust. Math., 18, Part 1, 1968, pages 31-62. 
G. F. Miller, Mathematical Tables: Volume 3. Tables of Generalized Expo- 
nential Integrals, Her Majesty’s Stationery Office, London, 1960. 
F. Oberhettinger, Tables of Fourier Transforms und Fourier Transforms of 
Distributions, Springer-Verlag, New York, 1990. 
A. P. Prudnikov, Yu. A. Brychov, and 0. I. Marichev, Integrals und Series, 
Volumes 1, 2, and 3, translated by N. M. Queen, Gordon and Breach, New 
York, 1990. 
A. D. Wheelon, Tables of Summable Series und Integrals Involving Bessel 
Functions, Holden-Day, Inc., San F’rancisco, 1968. 
IV 
Approximate Analytical 
Methods 
43. Asymptotic Expansions 
Applicable to Definite integrals that depend on a Parameter. 
Yields 
An asymptotic expansion. 
Idea 
When a Parameter in an integral tends to some limit, it may be possible 
to find an asymptotic expansion of the integral that is valid in that limit. 
Procedure 
There are several general asymptotic expansion theorems that can be 
used to determine the asymptotic nature of an integral; we enumerate only 
a few. 
Theorem (Bleistein and Handelsman [2], page 71): Define 
I ( X ) = Jabh( t ;X)f ( t )d t , where f(")(t) is continuous for n = 
O , l , . . . , N + 1, and f ( N + 2 ) ( t ) is piecewise continuous in the 
interval [U, b ] . If 
195 
196 IV Approximate Analytical Methods 
where the functions {a,(t)} are continuous on the interval 
[U, b] and the functions {&(z)} form an auxiliary asymptotic 
sequence as X -+ XO, then 
(43.1) 
n=O 
where the S,(X) are defined by 
S,(X) = ( - 1 y [f'"'(b)h'-"-l'(b; X) - f(n)(u)h(-n-l)(u; X)] . 
Special Case 1 
gral I (A) = s," h(Xt) f ( t ) d t has the asyrnptotic expansion 
Under appropriate smoothness and boundedness conditions, the inte- 
Special Case 2 
Under appropriate smoothness and boundedness conditions, the inte- 
gral I ( X ) = s," ezx t f ( t ) d t (which is a Special case of Special Case 1), has 
the asymptotic expansion 
Special Case 3 
Under appropriate smoothness and boundedness conditions, the inte- 
gral I (A) = s," e -x t f ( t ) d t (which is a Special case of Special Case 1)) has 
the asymptotic expansion 
Watson's Lemma (Bleistein and Handelsman [2], page 103, 
or Wong [ 5 ] , page 20): If f ( t ) is locally absolutely integrable 
on (0,00), as t + 00, f ( t ) = 0 ( e a t ) for some real number 
(Y, and, as t -+ 0+, f ( t ) N ~ n = O ~ m t a m , where Re(u,) 00 
43. Asymptotic Expansions 197 
increases monotonically to +oo as m -+ 00, and Re(a0) > -1, 
then, as X -+ 00, 
Theorem (Bleistein and Handelsman [2], page 120): Let 
h(t) and f ( t ) be sufficiently smooth functions on the infinite 
interval (0,oo) having the asymptotic forms 
m=O n = O 
m=O n=O 
with some conditionson the range of the Parameters appear- 
ing in the expansion. Let the Mellin transforms of h and f 
be denoted by M[h;z] and M [ f ; z ] (See the Notes). If some 
technical conditions are satisfied, then 
represents a finite asymptotic expansion as X -+ 00 with 
respect to the asymptotic sequence {X-.j (log X ) n j - m } . The 
expression in (43.2) represents a sum of the residues over all 
of the poles in a specific region of the complex plane. 
Bleistein and Handelsman [2] simplify t he expression appearing in 
(43.2) in five different cases, depending on the values of the Parameters. 
Example 
If J(X) = J: t x f ( t ) d t , then h(t; X) = t x f ( t ) and so 
~ X + n + l 
for n = 0, 1, . . . where we have Chosen the limit of integration in the repeated 
integrals of h to be Zero. From (43.1) this results in the following asymptotic 
expansion: 
N t X + n + l 
[ p ( b ) b ” + ” + ’ - f(”)(U)U~+”+’] (43.3) 
n=O 
whenever f satisfies the hypotheses of the theorem. If f were a polynomial, 
then the expansion in (43.3) would be the exact evaluation for J(X). 
198 
Notes 
IV Approximate Analytical Methods 
The Mellin transform of the function f( t ) is M[f; 21 = soW t"-'f(t) dt which 
can be interpreted as the ( z - 1)-st moment of f(t) . The so-called bilateral 
Laplace transform of g(t) is the Mellin transform of f(t) = g(- logt). 
Consider the integral I = Jabexf(")g(x)dx in the limit X -, 00. An asymp- 
totic expansion methodology for I is given by 
(A) the method of steepest descents (page 229) when f(x) is complex, 
(B) Laplace's method (page 221) when f(x) is real, 
(C) the method of stationary phase (page 226) when f(x) is purely imagi- 
In asymptotic formulas, it is important to describe fully the region in which 
a Parameter is tending to a limit. Error estimates should also be supplied 
with an asymptotic formula. For example, the complementary error function 
has the following expansions as x + 00: 
nary (no real component). 
00 
e - ~ ' z ( - i ) k ~ ~ + ~ (211 - 2k+l l)!! for -- 31r < argz < - 31r 4 4 ' erfc(x) N X 
k=O 
(43.4a-b) 
Note that these expansions both apply in the region of overlap, 7r/4 < 
argx < 37r/4. In this overlap region equation (43.4.a) has a large error 
while equation (43.4.b) has a small error. 
Wong [5] (page 22) has a generalized Watson's lemma: 
Define I ( A ) = JOWei' f(t)e-"dt, and assume that I(&) ex- 
ists. If f(t) N c,tam as t -, 0 along argt = y, where 
Re ao > 0 and Re a,+1 > Re um, then I ( X ) N crnqarn + 1) Xa,+l . 
Salvy [4] has created a package for automatically determining the asymptotic 
expansion of some classes of integrals, using the symbolic Computer language 
MAPLE. Salvy gives the following sample Outputs from his program: 
1 + tx X 
as x + 00. Here, y is Euler's constant. ' 
44. Asymptotic Expansions: Multiple Integrals 199 
Re ferences 
[i] 
[2] 
[3] 
C. M. Bender and S . A. Orszag, Advanced Mathematical Methods for Scien- 
tists und Engineers, McGraw-Hill, New York, 1978. 
N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, 
Dover Publications, Inc., New York, 1986. 
F. W. J. Olver, “Uniform, Exponentially Improved, Asymptotic Expansions 
for the Generalized Exponential Integral,” SIAM J. Math. Anal., 22, No. 5, 
September 1991, pages 1460-1474. 
B. Salvy, “Examples of Automatic Asymptotic Expansions,’’ SIGSAM Bul- 
letin, ACM, New York, 25, No. 2, April 1991, pages 4-17. 
R. Wong, Asymptotic Approximation of Integrals, Academic Press, New 
York, 1989. 
[4] 
[5] 
44. Asymptotic Expansions: 
Multiple Integrals 
Applicable to 
parame t er. 
Multidimensional definite integrals t hat depend on a 
Yields 
An asymptotic expansion. 
Idea 
When a Parameter in an integral tends to some limit, it may be possible 
to find an asymptotic expansion of the integral that is valid in that limit. 
Procedure 
It is difficult to state concisely very much about the different asymp- 
totic behaviors that are possible in multiple integrals. In this section we 
will only focus on the integral 
(44.1) 
when 4 is a real function and X is real with IXI + 00. Integrals of this 
form are known as integrals of Laplace type. (The other interesting case 
that occurs in applications is when X is purely imaginary with [X I + 00; 
this leads to integrals of Fourier type.) Here, D is some (not necessarily 
bounded) domain in n-dimensional x space. 
200 IV Approximate Analytical Met hods 
Laplace Type Integrals 
We presume that X is real and that D is a bounded simply connected 
domain. The boundary of D , denoted by r, is an (n - 1)-dimensional 
hypersurface. We assume it can be represented as 
where a takes values in the set P. We presume that 4, 90, and q ( a ) are 
sufficiently differentiable for what follows. 
There are now two cases, depending on where the maximum of 4 
appears in D. 
Maximum on the Boundary 
We presume that the maximum of 4 appears on the boundary D at 
the unique Point x = xo. 
Let N be the outward normal to r, and let dC be the differential 
element of “surface area” on r. Now define the gradient Operation, V = 
(axl,. . . , Ox,), the functions $ ( U ) = $(x(a)), Hj = gjV4/lV412, and 
gj+1 = V - Hj (for j = O , l , . . .). Then an exact representation of the 
integral in (44.1) is given by (see Bleistein and Handelsman [2], Page 332) 
M-1 
I ( X ) = - (--X)-j-’ 1 (Hj - N)eX4 dC + 9 gMex4 dx (44.2) 
j = O r 
for M = 2,3, . . . . 
The first terms in this expression are lower dimensional integrals for 
which asymptotic expressions may be found (recursively, if necessary). The 
last term can be bounded and will become the “error term.” 
For the particular case of n = 2, we can parameterize the boundary 
by I’ : ( q ( s ) , ~ ( s ) ) where s = 0 corresponds to the maximum at x = XO. 
The leading Order term in (44.2) (e.g., the j = 0 term) can then be written 
as 
(44.3) 
if the maximum of ik at s = 0 is simple (so that W’(0) < 0). If ~ ( 0 ) is the 
curvature of r at x = XO, then (44.3) can be further simplified to 
where the minus (plus) sign holds when r is convex (concave) at 
7 
X=Xo 
x = xo. 
(44.4) 
44. Asymptotic Expansions: Multiple Integrals 201 
Maximum Not on the Boundary 
We presume that the maximum of 4 appears in the interior of D at 
the unique Point x = XO. 
Near x = xo we can expand 4 in the form 4(x) - ~(xo) x i (x - 
xo)A(x - XO)~, where the matrix A = (a i j ) is defined by aij = 4zzz, (xo). 
Let Q be an orthogonal matrix that diagonalizes A , i.e., 
Al 0 
&TA&= ( ... A n ) * 
Then define the variable z by (x - XO) = QRzT where the matrix R = (ri j) 
is defined by rij = S i j lAil-1'2. Now the functions {hi} are Chosen so that 
6 = hi(z) = xi + o(IzI) (as IzI -+ 0) and E:', h: = 2 (~(xO) - ~ ( x ( z ) ) ) . 
d(xi, - - - 7 x n ) arid 
a(Ci, * * - 7 Cn) With these functions we define the Jacobian J ( C ) = 
then go(x(('>)J(C) = Go(C) = Go(0) + C - Ho. Then we have the recursive 
definitions Gj(C) = Gj(0) + C - Hj(6) and Gj+l(C) = V - Hj(C). (Note 
that there is an ambiguity in the {Hn}, this is not important.) Finally, the 
approximation to I ( X ) is given by (see Bleistein and Handelsman [2], Page 
335) 
(44.5) 1 I ( X ) - ex4(x0) A-jGj(O)Z[l] + X-MZ GM(^)] . r-l j = O 
where Z[Ic(<)] = JDIc(C)exp(-iXC.C) dC. Note that (44.5) is not an 
exact represent at ion, since M exponent ially small boundary integrals have 
been discarded. The leading Order term in (44.5) can be written as 
( F)n'2 9o(xo) . (44.6) ex4(X0! 
I(') - Jm- 
For the particular case of n = 2, the result in (44.6) can be written as 
Example 1 
J ex(z-Y2) dx dy where 
D is the unit circle. This integral has go = 1 and 4 = x - y2. In D, the 
maximum of 4 is at xo = ( 1 , O ) . Since this a boundary Point of D, (44.4) 
is the appropriate formula to use. The only nonzero terms that appear in 
Consider the two-dimensional integral J ( A ) = 
D 
(44.4) are: go(x0) = 1, ~(xo) = 1, &(xo) = 1, K(XO) = 1, 
1. Using these values resultsin the approximation J ( A ) - 
202 IV Approximate Analytical Methods 
Example 2 
Consider the two-dimensional integral K(X) = s 1 ex(2-22-Y2) dx dy 
where D is the unit circle. This integral has go = 1 and 4 = 2 - x 2 - y2. 
In D, the maximum of 4 is at xo = (0,O). Since this an interior Point of D, 
(44.7) is the appropriate formula to use. The only nonzero terms that 
appear in (44.7) are go(x0) = 1, ~(xo) = 2, ~zz(x~) = -2, and 4yy(x~) = 
-2. Using these values results in the approximation K(A) .rre2x/A. 
For this example, the integral can be computed exactly. We find 
D 
asX-,oo. 
e2’ - T- x 
Notes 
[l] 
[2] 
Bleistein and Handelsman [2] also describe asymptotic results for multidi- 
mensional Fourier integrals. 
Consider the integral I ( A ) = J J f ( ~ ) e z ’ ~ ( ~ ) dx, where X is a large Parameter. 
R2 
At caustic Points (also known as turning Points), defined by 
V ~ ( X O ) = 0 and det (”-> = 0, axiaxj 
the classical stationary phase techniques do not apply. For caustic Points 
where the Hessian determinant vanishes, but the Hessian is not identically 
the Zero matrix, there are several canonical forms of physical interest. These 
include the following possibilities for +(x, y): 
4(x,y) = x3 + y2 fold 
cusp 4(z,y) = x4 + y2 
4(z, y) = x5 + y2 
$(x,y) = x6 + y2 
swallowtai~ 
butterfly. 
For caustic Points where the Hessian is identically the Zero matrix, the 
canonical forms for #(x,y) (for Co-dimension less than 5) are 
3 4(x,y) = x - zy2 elliptic umbilic 
4(z, y) = x3 + y3 or x3 + xy2 hyperbolic umbilic 
parabolic umbilic. 4(x, Y) = x4 + xy2 
See Gorman and Wells [4] for details. 
Brüning and Heintze [3] derive an asymptotic expansion for the integral 
J g ($) xo logTxf(x) dx, as s + o+. [3] 
[ O J P 
204 
We define 
IV Approximate Analytical Met hods 
... 
b3 + - ...+ ak 
to be the k-th convergent of the continued fraction I . The continued 
fraction is said to converge if the sequence {cn} converges. 
In many cases, an integral may be written as a continued fraction. 
Partial convergents of the continued fraction then yield approximations to 
the original integral. 
Notes 
[l] Given {uk} and { b k } , define the seqiiences { p k ) and { q k } by the recurrence 
relations q k = b k q k - 1 + u k q k - 1 and p k = b k p k - 1 + u k p k - 1 for k = 1 , 2 , . . .. 
The initial values are given by p-1 = 1, PO = 0, q-1 = 0, and qo = 1. Then 
c k = plc/qk for k = 1 , 2 , . . .. Note that this relates continued fractions to 
recurrence relations. 
It might be easier to observe that the three-term recurrence relation, 
Yn + UnYn+1 + b,y,+l = 0, is formally equivalent to the continued fraction 
bn bn+i . . . . 
-an - -- 
Yn+i an+i- G + 2 - 
Yn - - 
[2] Rom a Taylor series in the form F ( z ) = cp",, d i ~ - ( ~ + l ) , a z-fraction, which 
eol fii- eil f 2 l . . ., may easily is a continued fraction of the form - - - - - - - - 
be constructed. By defining F k ( z ) = XEo di+kz-(Z+') - - c ~ ( z ) , where I2 I1 I2 I1 
we obtain the recurrence relations 
e j - i , k + i + fj,rc+i = fj,k + e j , k for e 
f j , k + l e j , k + l = e j , k f j + l , k for f 
with eO,k = 0 and f1 ,k = &+I/&. After determining the { e j , k , f j , k } , we 
find e j = e j , o and f j = f j , ~ for j 2 1, and eo = do. This is known as the 
QD (for quotient difference) algorithm. See van der Laan and Temme [3] 
for details. 
Using this algorithm and the asymptotic formula 
we can derive a continued fraction approximation to the erfc function: 
46. Integral Inequalities 205 
References 
[1] B. Char, "On Stieltjes Continued Fraction for the Gamma Functions," Math. 
of Comp., 34, 1980, pages 547-551. 
[2] D. Dijkstra, "A Continued Fraction Expansion for a Generalization of Daw-
son's Integral," Math. of Comp., 31, 1977, pages 503-510. 
[3] C. G. van der Laan and N. M. Temme, Calculation of Special Functions: 
The Gamma function, the Exponential Integrals and Error-like Functions, 
Centrum voor Wiskunde en Informatica, Amsterdam, 1984. 
46. Integral Inequalities 
Idea 
Some integrals may be easily bounded by known theorems. 
Procedure 
Given an integral that is to be bounded, a formula should be located 
that has the desired form. This is not a straightforward process. 
Example 
Suppose we would like to bound the integral I = ('~ dx. 
io 1 + x2 
If we write this integral as I = 101 f(x)g(x) dx, with f(x) = e-x and 
g(x) = 1/ VI + x 2 , then we note that both f and g are decreasing functions 
on the interval [0,1}. Hence, Tschebyscheff's inequality can be used to 
derive a lower bound (see the table at the end of this section for an exact 
statement of the inequality). We have 
To obtain an upper bound, we can use Holder's inequality with p = 
q = 2 (see the table at the end of this section for an exact statement of the 
inequality). We have 
206 IV Approximate Analyt ical Met hods 
- - 
Hence, we have found a fairly tight bound for I (that is, 0.556 < I < 0.584), 
without having to perform much computation. 
One Dimensional Inequalit ies - Named 
Carleman’s inequality (See Iyanaga and Kawada [9], Page 1422) 
lm exp (:[ log f(t) d t ) dz < f(z) dz 
when f(z) > 0. 
Cauchy-Schwartz-Bunyakowsky inequality (See Squire [19], Page 21) 
Equality occurs only when f(z) = kg(z), with k real. 
Hardy’s inequality (See Iyanaga and Kawada [9], Page 1422) 
when p > 1 and f(z) > 0. Equality is achieved only if f(z) = 0. 
Modified Hardy’s inequality (See Izumi and Izumi [8]) 
when m > 1, p > 1, and f(z) > 0. 
Hardy-Littlewood supremum theorem (See Hardy, Littlewood, and Polya [7], 
Page 298 (#398)) 
if k > 1 and f(z) is non-negative and integrable. 
46. Integral Inequalities 207 
[6] Hölder’s inequality (See Squire [19], page 21) 
/[f(z)9(4 dzl 5 ( / h ) l P dz) l lP ( Jlb19(z)i“ dz ) i/n 
when p and q are positive and l / p + l / q = 1. Equality occurs only when 
alf(z)lP = ,ßIg(z)Iq, where Q and ß are positive constants. 
Backward Hölder’s inequality (See Brown and Shepp [l]) [7] 
when f and g have compact Support, p and q are positive, and l / p+ i /q = 1. 
[8] Backward Hölder’s inequality (See Brown and Shepp [l]) 
/S;P [f(z - Y ) ~ ( Y ) ] da: 1 ( / l f ( z ) l p d z ) ‘ I p (/ls(r)lo dz) 
when f and g have compact Support, p and q are positive, and l / p+ l /q = 1. 
Minkowski’s inequality (See Squire [19], page 21) [9] 
(JI’If(4 +9(z)lpdz)1/p L ([lf(z)lpdz)l/y ( lb1901pdz)1 /p 
for p > 1. Equality occurs only when f(z) = kg(z), with k non-negative. 
[ l O ] Ostrowski inequality (See Gradshteyn and Ryzhik [6], page 1100) 
when f(z) is monotonic decreasing and f ( a ) f ( b ) 2 0. 
[ll] Tschebyscheff inequality (See Squire [19], page 22) 
when f(z) and g(z) are both increasing or both decreasing functions. 
[12] Tschebyscheff inequality (See Squire [19], page 22) 
when f(z) is an increasing function and g(z) is a decreasing function (or 
vice-versa) . 
[13] Wirtinger’s inequality (See Hardy, Littlewood, and Polya [7], page 185 (#257)) 
p(4 L p ) 2 ( z > dz 
Tf f(0) = f(r) = 0 and f’ is L2. Equality is obtained only if f(z) = C sin 2. 
[14] Generalized Wirtinger’s inequality (1 5 k < 00)’ (See Tananika [20]) 
208 IV Approximate Analyt ical Met hods 
[15] Young's inequality (See Hardy, Littlewood, and Polya [7], page 111 (#156)) 
ab 5 l a f ( x ) dz + L b f - ' ( z ) dx 
when f(z) is continuous, strictly monotone increasing in z 2 0, f(0) = 0, 
a 2 0, and b 2 0. Equality occurs only if b = f(a). 
One Dimensional Inequalit ies: 
Arbitrary Intervals - Unnamed 
If p ( z ) > 0 and f p ( z ) dx = 1 (See Hardy, Littlewood, and Polya [7], page 
137 (#184)), then (unless f is a constant) 
If 0 < T < s, p(x) > 0, and f p ( x ) d z = 1 (See Hardy, Littlewood, and 
Polya [7], page 143 (#192)), then (unless f is a constant) 
If 0 < U 5 f(x) 5 A < 00 and 0 < b 5 g ( x ) 
Littlewood, and Polya [7], page 166 (#230)), then 
5 B < 00 (See Hardy, 
If U , b, <Y, ß are positive and f(x) is an increasingpositive function (See 
Hardy, Littlewood, and Polya [7], page 297 (#397)), then 
If 1 5 T < p and f(x) and g ( x ) are positive functions in L p (See Potze and 
Urbach [21]), then 
where C,.,p 2 0. 
46. Integral Inequalities 209 
One Dimensional Inequalit ies: 
Finite Intervals - Unnamed 
If U 2 0, b 2 0, U # 1, f(x) is non-negative and decreasing, and f(x) # C 
(See Hardy, Littlewood, and Polya [7], page 166 (#229)), then 
( L i x a + b f d x ) 2 5 [l - ( u + b + l - * )'3 ( J / i x 2 a f d x ) ( L i x 2 b f d x ) . 
If f(x) has period 2n, s,"" f dx = 0, f' is L2, and f(x) # A sinz + B COS x 
(See Hardy, Littlewood, and Polya [7], page 185 (#258)), then 
L 2 " f 2 ( x ) d z < 1 2 " ( f ' ( x ) ) 2 dx . 
If 0 5 f' 
page 298 (#400)), then 
1, 0 5 g ( x ) < x, and k > 1 (See Hardy, Littlewood, and Polya [7], 
If 0 5 f ' 5 1 and 0 5 g(x) < x (See Hardy, Littlewood, and Polya [7], page 
298 (#400)), then 
One Dimensional 
. . 
Infinite Intervals - Unnamed 
[25] If m > 1, n > -1, f is positive (see Hardy, Littlewood, and Polya [7], page 
165 (#226)), then 
l w x n f " ( x ) dx 
Equality occurs only when f = B exp ( - C Z ( ~ + ~ ) / ( ~ - ' ) ), where B 2 0 and 
c > 0. 
[26] If a 2 0, b 2 0, U # b, and f is non-negative and decreasing (See Hardy, Lit- 
c in (070 
0 in (t700) tlewood, and Polya [7], page 166 (#228)), then (unless f(x) = 
with C > 0) 
210 IV Approximate Analytical Met hods 
Tf f and f" are in L2[0, 001 (See Hardy, Littlewood, and Polya [7], page 187 
(#259)), then 
Equality occurs only when f(z) = AeFBXI2 sin (Bzsin $ - $) 
If f and f" are in L2 [0,003 (See Hardy, Littlewood, and Polya [7], page 188 
(#260)), then 
JOm (f2(.) - ( f W 2 + ( f W 2 ) dz 2 0. 
Equality occurs only when f(z) = Ae-Bx/2 sin (Bxsin $ - $) 
If f and f" are in L2[-m,0o] (See Hardy, Littlewood, and Polya [7], page 
193 (#261)), then (unless f(x) = 0) 
If p > 1 and f(x) 2 0 (See Hardy, Littlewood, and Polya [7], page 240 
(#327)), then (unless f(z) = 0) 
If p > 1, 0 I a < l /p, and p 5 q 5 p/ ( l - ap) (See Hardy, Littlewood, and 
Polya [7], page 298 (#402)), then 
This result is also true if a 2 l/p, p > 1, and p 5 q. In both cases 
K = K(p,q,a) > 0. 
[32] Under some continuity requirements, with a > 0 (See Mingarelli [ lO] ) 
46. Integral Inequalit ies 211 
Two Dimensional Inequalit ies 
[33] I f p > 1 , q > 1 , p - 1 + q - 1 ~ 1 , X = 2 - p - 1 - q - 1 , h < 1 - p - 1 , k < l - q - l , 
h + k 2 0, and h + k > 0 if p - l + q-l = 1 (See Hardy, Littlewood, and 
Polya [7], page 298 (#401)), then 
Here K = K(p, q, h, k) > 0. 
[34] If f(x), g(x) and h(x) are non-negative, and f*(x), g*(x) and h*(x) are the 
equi-measurable symmetrically decreasing functions (See Hardy, Littlewood, 
and Polya [7], page 279, (#379)), then 
[35] If p > 1, p‘ = p / ( p - 1), ~ o O O f*(x)dx I F and Jomgp’(x)dx I G (See 
Hardy, Littlewood, and Polya [7], page 226 (#316)), then (unless f(x) = 0 
or g(x) = 0) 
0 t her Inequalit ies 
[36] If the function f,g, . . . , h are linearly independent functions (i.e., there do 
not exist constants A, B , . . . , C, some not equal to zero, such that A f + B g + 
- - -+Ch = 0) (See Hardy, Littlewood, and Polya [7], page 134 (#182)), then 
J f2 (4 J f(49(4 dx * * * J f(z)h(z) dx 
> 0. 
J h(x)f(z) dx J h ( z ) g ( x ) dz . . . J h2(x) dx 
Notes 
[l] In this section, when no further explanation is given, functions with upper 
case letters are assumed to be integrals of functions with lower case letters. 
For example, F ( x ) = Joz f(x) dx and G(z) = soz g(z)dx. Also, all the 
integrals in this section are assumed to exist. 
If f(z) is a real continuously differentiable function that satisfies the bound- 
edness constraints x21f(x)I2 dx < 00 and 1:- lf’(x)I2 dx < 00, then 
for x 2 0 we have 
[2] 
212 IV Approximate Anaiytical Methods 
This, in turn, can be used to derive the inequality 
This last inequality is known as Heisenberg’s uncertainty principle in quan- 
t um mechanics . 
Evans et al. [3] contains a complete analysis of the inequality [3] 
[4] Pachpatte [18] derives generalizations of the inequalities 
with suitable constraints on f, m, and p . (Here, F, is related to the integral 
The inequality ss l f I 2 d z d y 5 - (s, I f l d)z l ) for functions f holomor- 
phic in G U I? is referenced in Gamelin and Khavinson [5 ] . 
Gronwalls’ inequality states (See Gradshteyn and Ryzhik [6], page 1127): 
of f.) 
1 2 
[5] 4lr G 
[SI 
Theorem: Let the three piecewise continuous, nonnegative 
functions {U, U , 20) be defined in the interval [0, U ] and satisfy 
the inequality 
except at Points of discontinuity of the functions. 
except at these Same Points, 
Then, 
w ( t ) 5 ~ ( t ) + 1’ u ( r ) w ( ~ ) exp ([ v(o) do) d r . 
46. 
[71 
Integral Inequalit ies 213 
Opial [12] showed that 1; If(x)l f'(z) dx I ah l;(f'(~))~ dx, with certain 
conditions on f . A comprehensive survey of Opial-type inequalities may be 
found in Mitrinovib [ll]. Yang [22] proved the generalization 
Theorem: If f(s, t ) , fs, and fst are continuous functions on 
[a,b] x [c,d] and if f ( a , t ) = f ( b , t ) = fs(s,c) = f S ( s , d ) = 0 
for U 5 s I b and c I t I d , then 
Two other generalizations of Opial's inequality are in Pachpatte [16]. 
Theorem: Suppose the functions p , q are positive and con- 
tinuous on A = [.,XI x [ c , Y ] . Let f = f ( s , t ) , fs, fst be 
continuous functions on A with f ( a , t ) = f s ( s , c ) = 0 for 
U 5 s 5 X and c I t 5 Y. If m and n are positive integers, 
with m + n > 1, then 
One of these generalizations is (the other is similar): 
1, JlYPIfl" l f d dtds I ~ ( X , Y , m , n ) Lx Ly q lfst I m + n d t ds (46.1) 
where K ( X , Y, m, n ) is a finite constant that depends on the 
functions p and q. If m < 0, n > 0, and m + n > 1, then 
(46.1) holds with 5 replaced with 2. 
[8] Assume that f ( t ) and 4( t ) are nonnegative and measurable on R+, and that 
both U and b are in the range (0,oo). Define @(x) = Joz 4( t ) d t , F L ( ~ ) = 
Joz f ( t )4 ( t )d t , Fv(x) = JZw f ( t ) + ( t ) d t , and M = (p/Ic - 11)". Then (See 
Copson [2]): 
i f p > l , c > l 
if 0 < p 5 1, c > 1 
and @ ( x ) -t 0 as x -t 00 
i f p > l , c < l 
if 0 < p 5 1, c < 1 
214 IV Approximate Analytical Met hods 
[9] The HELP (Hardy, Everitt, Littlewood, Polya) inequalities are of the form 
(See Evans and Everitt [4]) 
wherep, q, and 20 are real-valued functions on [U, b] (with -00 < U < b 5 oo), 
and M [ - ] denotes the second Order differential expression M [ f ] = -(pf’)’ + 
q f . There are some technical conditions on p , q, and w. 
Re ferences 
G. Brown and L. A. Shepp, Backward Hölder’s Inequality,” Problem 
number E 3370 in Amer. Math. Monthly, 98, No. 7, August-September 
1991, pages 650-652. 
E. T. Copson, “Some Integral Inequalities,” Proc. Roy. SOC. Edinburgh, 75A, 
No. 13, 1975/76, pages 157-164. 
W. D. Evans, W. N. Everitt, W. K. Hayman, and S. Ruscheweyh, “On a 
Class of Integral Inequalities of Hardy-Littlewood Type,” J . Analyse Math., 
46, 1986, pages 118-147. 
W. D. Evans and W. N. Everitt, “HELP Inequalities for Limit-Circle and 
Regular Problems,” Proc. R. SOC. London A, 1991, 432, pages 367-390. 
T. W. Gamelin and D. Khavinson, “The Isoperimetric Inequality and Ra- 
tional Approximation,” Amer. Math. Monthly, January 1989, Page 22. 
I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, und Products, 
Academic Press, New York, 1980. 
H. Hardy, J. E. Littlewood, G. Polya, Inequalities, Cambridge Mathematical 
Library, Second Edition, New York, 1988. 
M. Izumi and S. Izumi, “On Some Inequalities for Fourier Series,” J. Anal. 
Math., 21, 1968, pages 277-291. 
S. Iyanaga and Y . Kawada, Encyclopedic Dictionary of Mathematics, MIT 
Press, Cambridge, MA, 1980. 
A. B. Mingarelli, Note on Some Differential Inequalities,” Bull. Inst. 
Math. Acad. Sinica, 14, No. 3, 1986, pages 287-288. 
D. S. MitrinoviC, Analytic Inequalities, Springer-Verlag,New York, 1970. 
Z. Opial, “Sur une inegalite,” Ann. Polon. Math., 8, 1960, pages 29-32. 
K. Ostaszewski and J. Sochacki, “Gronwall’s Inequality and the Henstock 
Integral,’’ J. Math. Anal. Appl., 127, 1987, pages 370-374. 
B. G. Pachpatte, “On Opial-Type Integral Inequalities,” J. Math. Anal. 
Appl., 120, No. 2, 1986, pages 547-556. 
B. G. Pachpatte, “On Some Variants of Hardy’s Inequality,” J. Math. Anal. 
Appl., 124, 1987, pages 495-501. 
B. G. Pachpatte, “On Two Independent Variable Opial-Type Integral In- 
equalities,” J. Math. Anal. Appl., 125, 1987, pages 47-57. 
B. G. Pachpatte, “On Some New Integral Inequalities in Two Independent 
Variables,’’ J. Math. Anal. Appl., 129, No. 2, 1988, pages 375-382. 
B. G. Pachpatte, “On Some Integral Inequalities Similar to Hardy’s Inequal- 
itv.” J. Math. Anal. Avvl.. 129. 1988. Dages 596-606. 
Integration by Parts 215 
W. Squire, Integration for Engineers and Scientists, American Elsevier Pub- 
lishing Company, New York, 1970. 
A. A. Tananika, “A Generalization of Wirtinger’s Inequality,” Diflerentsia2’- 
nye Uravneniya, 22, No. 6, 1986, pages 1074-1076. 
W. Potze and H. P. Urbach, “An Inequality in LP,” Appl . Math. Lett., 3, 
No. 3, 1990, pages 95-96. 
G. S. Yang, “Inequality of Opial-type in Two Variables,” Tamkang J. Math., 
13, 1982, pages 255-259. 
47. Integration by Parts 
Applicable to A Single integral. 
Yields 
Idea 
be obtained. 
An asymptotic expansion of the integral. 
By using integration by parts an asymptotic expansion may sometimes 
Procedure 
Repeatedly using the process of integration by parts (See Page 161) 
often allows an asymptotic expansion to be obtained. The remainder term 
is needed to determine the error at any Stage of the approximation. 
Several theorems are available that can be used to state an asymptotic 
expansion of an integral immediately, See the Notes. 
Example 1 
The exponential integral is defiried by 
We make t he ident ificat ion 
integration by parts formula, 
El(x) = lrn d t . t (47.1) 
( d v = e- td t , U = t-’), and then use the s u d v = U V I - s v d u , on (47.1) to obtain 
(47.2) 
Making the identification ( d v = e- tdt , U = t-2) in (47.2), and using 
integration by parts again, results in 
‘-“ + lm dt . e-“ El(X) = - - - 
x x2 
216 IV Approximate Analytical Met hods 
Integrating by parts a total of N times results in 
(47.3) N - 1 
= (e-. 7) ( -l)nn! + ( ( - l ) " N ! L m G d t ) . 
n=O 
- i ) n n Integrating by parts infinitely many times results in e-" E,"=, G. 
This series diverges for all values of x (since the absolute value of the 
ratio of successive terms in the sum is n / x , which increases as n increases), 
and so is not a good representation of E l ( x ) . 
However, it is not hard to bound the remainder term R N ( x ) . We have 
- (d=e-"NN) e-" xN+l 
N 
where we have used Stirling's approximation for N ! , which is asymptotically 
valid for large values of N . F'rom this rough approximation, we conclude 
t hat, for some values of x , smaller values of N may give a smaller remainder 
than larger values of N . 
Example 2 
The complementary error function is defined by the integral erfc(z) = sxw e-t2 d t . By repeated integration by parts, we can obtain the asymp- 
totic expansion 
1 1 3 
k (1)(3) * ' ' (2k - 1) - --E(-') 2k+lX2k+1 ' 
J?r k=O 
Once again, this asymptotic series diverges for all values of x . However, for 
a fixed number of terms, the approximation becomes better as x increases. 
A numerical illustration of this asymptotic expansion is in Table 47. 
47. Integration by Parts 217 
Table 47. A numerical comparison of the complementary error function with 
the first term and first two terms in its asymptotic expansion. 
- 2 2 
erfc(x) 
x f i 
x = 1 0.15730 0.41510 0.20755 
x = 2 0.00468 0.00517 0.00452 
x = 3 0.0000221 0.0000232 0.0000219 
Notes 
[l] The approximation in (47.3) indicates a common trait of asymptotic se- 
quences: For fixed x, the approximation gets worse as N increases, for 
fixed N , the approximation gets better as x increases. 
Note that, from a numerical Point of view, the integral in (47.2) is more 
rapidly converging than the integral in (47.1). 
Wong [4] presents the following example where integration by parts does not 
lead to an asymptotic expansion. The integral I(x) = 
can be integrated by parts to obtain 
[2] 
[3] 
dt l- (1 + t y 3 ( x + t ) 
(47.4) 
3N-l 
where SN(X) = ( N - 'I! 1" (' 4- t ) N - 4 / 3 d t . Approximations aris- 
ing from (47.4) are not useful, since I(x) is positive but every term in (47.4) 
is negative. 
Bleistein and Handelsman [2] have several general results about integration 
by parts. 
(A) Consider the integral I ( X ) = Jab h(t; X ) f ( t ; X) dt . Assuming sufficient 
continuity, we have I ( X ) = xn=O Sn(X) + RN(X) (See Bleistein and 
Handelsman [2], Theorem 3.1) where 
2 . 5 - ( 3 N - 4 ) < (x+ t )N 
[4] 
N 
where g ( n ) denotes the n-th derivative of g if n is positive, and it denotes 
the Jnl-th integral of g if n is negative. If 
asymptotic sequence as X -+ XO, and if 
218 IV Approximate Analyt ical Met hods 
(B) 
0 
0 
0 
0 
where the {an} are 
C;=,S~(X). AS an 
if 0 5 U < b, then we 
continuous, then as X -, XO we have I ( X ) - 
example, if f is sufficiently differentiable, and 
have 
a s X - - + m . 
Consider the integral I ( X ) = s," h(Xt)f(t) d t . Assuming 
sufficient continuity for f ; 
b - U is finite; 
Ih(-")(Xt)l 5 an(t)&(X), where the {an} are continuous, 
then, as X + 00 (See Bleistein and Handelsman [2], Theorem 3.2) 
X-'4n+l(X) = 0(4n(X)) as X + 0; 
As an example, we have the result (if f is sufficiently smooth) 
asX-,m. 
References 
[l] 
[2] 
[3] 
[4] 
C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scien- 
tists und Engineers, McGraw-Hill, New York, 1978 
N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, 
Dover Publications, Inc., New York, 1986, Chapter 3, pages 69-101. 
A. Erdelyi, Asymptotic Expansions, Dover Publications, Inc., New York, 
1956. 
R. Wong, Asymptotic Approximation of Integrals, Academic Press, New 
York, 1989, pages 14-19. 
48. Interval Analysis 
Applicable to 
expressions. 
Yields 
Ordinary integrals, or integrals containing interval 
An analytical approximation with an exact bound on the error. 
48. Interval Analysis 219 
Idea 
In interval analysis, quantities are defined by intervals with maximum 
and minimum values indicated by the endpoints. Definite integrals can 
often be approximated by an interval; intervals are better than ordinary 
numerical approximations since an exact bound on the error is obtained. 
Procedure 
We use the interval notation [U, b] to indicate some number between the 
values of U and b. We will allow coefficients of polynomials to be intervals. 
For example, t he int erval polynomial 
Q ( x ) = 1 + [2,3]x2 + [-1,4]x3, (48.1) 
evaluated at the Point x = y, means that 
min 1 + qy2 + <y3 I Q(y) I max 1 + qy2 + <y3. 
2 5 V 5 3 2 5 ~ 1 3 
There exists an algebra of interval polynomials. For example 
(48.2) 
-15614 -1sc54 
(x + [2, 3]x3) + ([I, 212 + [I, 4]x3) = [2,3]2 + [3, 7]x3, 
([1,3] + [ - 1 , 2 ] ~ ) ~ = [1,9] + [-6,1212 + [-2, 4]x2. 
If P (y ) and Q(y) are interval polynomials, then at any Point y we can 
write P ( y ) E [PL, Pu], Q(y) E [QL, &U]. We say that P(x ) contains Q(z) 
on some interval [c, d] if PL 5 QL, Qu 5 Pu for all y E [c, 4. This is 
denoted by Q(z) C P(x) . 
We now use capital letters to denote intervals; i.e., F ( X ) denotes the 
interval [FL, FH] where FL = minzEX f(z) and FH = rnax,Ex f(x). If we 
define X i n ) by 
1 1: n - u) ,a + -(z - U) , i = 1 ,2 , . . . ,n, (48.3) 
then, if f(z) is sufficiently smooth, 
lz f ( t ) d t c 2 F ( X j n ) ) (Y) . 
i 
(48.4) 
Define Q to the right-hand side of (48.4). The width of the interval Q, 
w(Q), can be shown to satisfy 
K ( x - u ) ~ 
n 4Q) I (48.5) 
where K is a positive constantindependent of n. 
The quadrature formula in (48.4) is essentially a first Order integration 
formula for J f ( t ) d t . Higher Order formulae are also available. See, for 
example, Corliss and Ra11 [4]. 
220 IV Approximate Analytical Methods 
Example 1 
This example illustrates the use of (48.4). Consider the integral I = 
sin7rx2 dx. Using (48.4) with n = 2, we have 
(48.6) 
where F ( X ) = sin(7rX2). The expression in (48.6) can be evaluated to 
yield 
= [o, 3 + + [o, 13 + 
= [o, i] . 
Example 2 
It is straightforward to Show that 
Consider the integral I = J!l f(x) dx, where f(x) = 1 + [1,2]x + x2. 
[i + 22 + x2, 1 + x + x 2 ] 
1 + x + x2, 1 + 22 + x2] 
for -1 5 x 5 0, 
for o 5 x 5 1. f (x> = { [ 
Hence, we have 
1 0 1 
1 + 22 + x 2 + J, 1 + x + x q l 1 + x +x2 + J, 1 + 22 +x2] 
I = [L1 
= (+, !). 
Notes 
[l] The techniques presented in this section can be evaluated numerically. In- 
terval arithmetic packages are available in Algol (See Guenther and Mar- 
quardt [ 5 ] ) , Pascal (See WOB von Gudenberg [lO]), and FORTRAN (See 
ACFUTH [l], but See also Kahan and LeBlanc [6]). 
Corliss and Ra11 [4] describe an interval analysis program that evaluates 
integrals by implementing Newton-Cotes rules, Gauss rules, and Taylor 
series. They also consider Problems in which the limits of integration are 
intervals. 
[2] 
49. Laplace’s Method 
References 
221 
ACRITH High Accuracy Subroutine Library: General Information Manual, 
IBM publication # GC33-6163, Yorktown Heights, NY, 1985. 
0. Caprani, K. Madsen, and L. B. Rall, “Integration of Rational Functions,” 
SIAM J. Math. Anal., 12, 1981, pages 321-341. 
G . Corliss and G. Krenz, “Indefinite Integration with Validation,” A C M 
Trans. Math. Software, 15, No. 4, December 1989, pages 375-393. 
G. F. Corliss and L. B. Rall, “Adaptive, Self-Validating Numerical Quadra- 
ture,” SIAM J. Sci. Stat. Comput., 8, No. 5, 1987, pages 831-847. 
G. Guenther and G. Marquardt, “A Programming System for Interval Arith- 
metic,” in K. Nickel (ed.), Interval Mathematics 1980, Academic Press, New 
York, 1980, pages 355-366. 
W. Kahan and E. LeBlanc, “Anomalies in the IBM ACRITH Package,” 
IEEE Proc. 7th Symp. on Computer Arithmetic, 1985. 
R. E. Moore, Interval Analysis, Prentice-Hall Inc., Englewood Cliffs, NJ, 
1966, Chapter 8, pages 70-80. 
L. B. Rall, “Integration of Rational Functions 11. The Finite Case,” S I A M 
J. Math. Anal., 13, 1982, pages 690-697. 
J. M. Yohe, “Software for Interval Arithmetic: A Reasonable Portable Pack- 
age,” A C M Trans. Math. Software, 5, No. 1, March 1979, pages 50-63. 
J. Wolff von Gudenberg, Floating-Point Computation in P A S C A L-SC with 
Verified Results, in B. Buchberger and B. F. Caviness (eds.), EUROCAL 
’85, Springer-Verlag, New York, 1985, pages 322-324. 
49. Laplace’s Method 
Applicable to 
is a real-valued function. 
Integrals of the form I ( X ) = J: g(z)e’f(”) dz, where f(z) 
Yields 
An asymptotic approximation when X > 1. 
Idea 
those Points where f(z) is a local maximum. 
For X + 00 the value of I ( X ) is dominated by the contributions at 
222 IV Approximate Anaiytical Methods 
Procedure 
for fixed X. This 
term will have a stationary Point, (i.e., a local maximum or minimum) 
when de’f(x)/dz = 0, or f’(z) = 0. The behavior of I ( X ) is dominated at 
the local maximums of f ; Points where f’(x) = 0 and (usually) f”(z) < 0. 
If the stationary Point zi is an interior Point (i.e., U < zi < b) then 
I ( X ) may be approximated, in the neighborhood of this Point, as 
Given the integral I ( X ) , consider the term 
w I’*+‘ exp [X (f(zi> + (z - z,)2f”(zi) 2 +. . . ) I 
Xi -& 
(49.1) 
which is valid as X + 00, when f”(zi) < 0. If f”(zi) > 0, then the Point 
zi is a local minimum of f, not a local maximum, and this Point does 
not contribute to leading Order. (For the case f”(zi) = 0, see the Notes, 
below . ) 
For each Point where f has a local maximum there will be a term in 
the form of (49.1). To find the asymptotic approximation to I ( X ) , these 
terms must be summed up. 
For the stationary boundary Points (i.e., those stationary Points that 
are on the boundary of the domain, either zi = U or zi = b) there is a 
term in the form of (49.1), but with half the magnitude (if f”( ) is negative 
at that boundary Point). This is because the integral in the fourth line of 
(49.1) becomes an integral from 0 to 00 or -00 and not from -00 to 00. 
If either of the boundary Points is not a stationary Point, (i.e., f’(u) # 
0 or f ’ ( b ) # 0), then these boundary Points Points contribute 
(49.2) 
to the sum forming the the asymptotic approximation of I ( X ) . The contri- 
butions from non-stationary boundary Points will always be asymptotically 
49. Laplace's Method 223 
smaller than the contribution from the Points at which f is a local max- 
imum. Hence, if the region of integration contains any Points at which f 
is a local maximum, and if only the leading Order behavior is desired, then 
the non-stationary boundary Points can be ignored. 
Example 1 
For the integral J(X) = fto e-xcOsz dx we identify f (x) = - cosx, 
g(x) = 1, U = 0 and b = 10. The stationary Points are where f'(x) = 
sinx = 0, or x = (0, 7r, 27r, 37r, 47r,. . .}. We are only interested in those 
stationary Points in the range of integration, that is x = 0, x = 7r, x = 27r, 
and x = 37r. Including the boundary Points x = 0 and x = 10 we have 
four Points that can potentially contribute to the leading Order term in the 
asymptotic expansion. 
x = o 
x=7r 
x = 27r 
x = 37r 
x = 10 
This is a stationary boundary Point. However, since f"(0) = 1 > 
0, this Point does not contribute to leading Order. 
This is an interior stationary Point. Since f"(7r) = -1 < 0, this 
Point contributes a term of the form in (49.1): 
This is an interior stationary Point. However, since f"(27r) = 1 > 
0, this Point does not contribute to leading Order. 
This is an interior stationary Point that is a local maximum since 
f"(37r) = -1 < 0. Hence, this Point contributes a term of the 
This is a non-stationary boundary Point. Hence, this point con- 
We can combine all of the leading Order contributions we have to find 
, -x CO9 10 
X sin 10 
, -x CO9 10 
X sin 10 ' 
- - J(X) N I , + I ~ , + I ~ o = fi -e x + G e A - 
However, since we have only kept the leading Order term in the asymptotic 
expansion near each Point, we can only 
final answer (assuming no cancellation 
result is t herefore: 
keep the leading Order term in the 
of terms has occurred). Our final 
as X+m. 
224 IV Approximate Analytical Methods 
Example 2 
Consider the integral J(X) = Ji eXz2 dx, so that f(x) = x2 and g(x) = 
1. In this case the stationary Points are given by f'(x) = 2x = 0, or x = 0. 
The Point x = 0 is a stationary boundary Point, but it does not contribute 
to leading Order since f"(0) = 2 > 0. The Point x = 1 is a boundary Point 
and the leading Order asymptotic approximation is given by (49.2): 
Example 3 
Consider the integral K(X) = J: e-xz2 dx, so that f(x) = -x2 and 
g(x) = 1. In this case the stationary Points are given by f'(x) = -22 = 0, 
or x = 0. The Point x = 0 is a stationary boundary Point, and it contributes 
to leading Order since f"(0) = -2 < 0. The leading Order asymptotic 
approximation is given by (49.1) with a factor of $ (since x = 0 is a 
boundary Point): 
For this integral, we recognize that K(X) = 
the error function. Use of the asymptotic expansion of the error function 
for large arguments also results in (49.3). 
Example 4 
An integral representation of the gamma function, for X > 0, is 
00 00 
r ( X ) = 1 xx-'e-zdx = 1 2-l e -z e xlOgz dx. (49.4) 
If X is an integer, then r ( X ) = (X - l)! (see page 163). 
Rom (49.4) we have f(x) = - logx, but f(x) has no finite stationary 
Point about which to apply our above expansions. The Change of variable 
x = Xy transforms (49.4) to 
(49.5) 
Now f (y ) = -y + logy and g(y) = y-l. The minimum of f ( y ) is at 
f '(y)= 0, or y = 1. The value y = 1 is an interior stationary Point of 
(49.5), so we have (from (49.1)) 
From (49.6), and a little manipulation, we obtain the leading term in 
Stirling's approximation to the factorial: r(n + 1) = n! N &nne-n. 
49. Laplace’s Met hod 225 
Example 5 
exz-(z-l)lOgz dx, for X > 1. If we 
make the obvious identification, f(x) = x, then the region of maximum 
contribution will be around x = 00. To determine this contribution, some 
re-scaling of the Problem is required. 
Looking at the whole integrand, the stationary point is given by 
Consider the integral I ( X ) = 
x - 1 d dx (exz-(z-l) logz ) = o , or x = - X + logx. 
As suggested above, the stationary point occurs at a large value of x. If 
x is large then, approximately, the stationary point is given by x = ex-’. 
Making the Change of variable t = x/ex-’, we are led to consider 
where J(C) = som teC(t-t logt) dt. Since we Want X > 1, this corresponds to 
C > 1. Now it is a simple matter to show that J(C) N e C d m (since the 
only stationary point is at t = 1). Hence, we obtain our final answer: 
Notes 
[l] Laplace’s method is an application of the method of steepest descents, See 
page 229. In the method of steepest descents, the function f(x) can be 
complex valued. In the method of stationary phase (page 226)) the function 
f(z) is purely imaginary. 
If more terms are kept in (49.1), then we obtain the approximation [2] 
where all the functions are evaluated at z = xi (See Bender and Orszag [l], 
page 273). If the asymptotic expansion in Example 4 were continued to 
higher Order, then we would find 
This yields a better approximation to the factorial function than the one- 
term Stirling’s approximation. 
Watson’s lemma (page 197) applies to integrals of the form s,” eAgF(y) dy. 
By an appropriate Change of variable, I ( X ) can be changed to this form. 
For example, we can use the transformation y = f(x) and then (assuming 
monotonicity) A = f (a) , B = f (b) , F (y) = g(z)/ f’ (2). 
[3] 
226 IV Approximate Analytical Methods 
[4] If the second derivative vanishes at an interior stationary Point, so that the 
leading Order behavior at a stationary Points is given by 
for x near x i , then the first term in the asymptotic approximation of I ( X ) 
becomes (assuming that n is even, and f ( ” ) ( x i ) < 0, both of which are 
required for xi to be a local maximum): 
(49.7) 
[5] 
[6] 
(71 
Multidimensional analogues of this technique are described on page 199. 
Skinner [3] considers uniform approximations for integrals of the form 
eXh( t ) L” e -xh(z )g (x )xa- l dx as x -+ 00. 
Temme [4] considers uniform approximations for integrals of the form 
( i /r (x) ) Jam t x - l e - z t f ( t ) d t as x -, 00. 
C. M. Bender and S. A. Orszag, Adwanced Mathematical Methods for Scien- 
tists und Engineers, McGraw-Hill, New York, 1978. 
N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, 
Dover Publications, Inc., New York, 1986. 
L. A. Skinner, “Uniformly Valid Composite Expansions for Laplace Inte- 
grals,” SIAM J. Math. Anal., 19, No. 4, July 1988, pages 918-925. 
N. M. Temme, (‘Incomplete Laplace Integrals: Uniform Asymptotic Expan- 
sion with Application to the Incomplete Beta Function,” SIAM J. Math. 
Anal., 18, No. 6, November 1987, pages 1638-1663. 
References 
[l] 
[2] 
[3] 
[4] 
50. Stationary Phase 
Applicable to 
f(z) is a real-valued function. 
Yields 
Integrals of the form I ( X ) = s,” g ( z ) e i x f ( ” ) dz, where 
An asymptotic approximation when A > 1. 
Idea 
those Points where f(z) is a local minimum. 
For X + 00 the value of I ( X ) is dominated by the contributions at 
50. Stationary Phase 227 
Procedure 
The Riemann-Lebesgue lemma states that lim h(t)eixt dt = 0, 
provided that s,” Ih(t)l dt exists. In simple terms, if the integrand is highly 
oscillatory, then the value of the integral is “small.” 
x-Ca 
Now consider the integral 
I ( X ) = e i x f ( z ) g ( x ) dx. (50.1) 
Through an appropriate Change of variables, it can be shown (in non- 
degenerate cases) that I (X) + 0 as X + 00. The maximum contributions 
to (50.1) will come from regions where the integrand is less oscillatory. 
These regions are specified by the stationary Points of the integrand, that 
is, where f’(x) = 0. 
Let the stationary Points in the interval [a,b] be { c i } . Following a 
derivation similar to that given for Laplace’s method (see Page 221), we 
J / ) 
find that the leading Order contribution to I ( X ) , due to the 
Point c, is (assuming that f(c) # 0, f”(c) # 0, and g(c) # 0): 
stationary 
(50.2) 
where sgn denotes the signum (or sign) function. 
The leading Order asymptotic behavior of I (X) is then given by Ci Ic i . 
It is difficult to obtain a better approximation than just the leading Order 
approximation, because it requires delicate estimation of integrals. If a 
higher Order approximation is desired, then the method of steepest descents 
(see Page 229) should be used. 
Example 
representation (see Abramowitz and Stegun [l], 9.1.22.b) 
The Bessel function Jn(x), for integral values of n, has the integral 
1 f’ 
- 2’, 
1’ e f i n t e F i X sin t dt . (50.3) 
Each integral in the sum has the form of (50.1), with f*(t) = ~ s i n t and 
g * ( t ) = e f i n t . On the interval [0,7r] the only stationary Point for f* is at 
t = 7r/2. Hence, we find fk (5) = ~1 and fz (5) = ~ 1 . Therefore, (50.2) 
can be evaluated to yield 
228 IV Approximate Analytical Met hods 
for X > 1. 
Notes 
[l] Determining the asymptotic behavior of an integral by plugging into the 
above formulas is a dangerous approach. A simple example where this naive 
approach could go wrong is with the integral I = s-", ei(3Xt2-3t3) dt . U se 
of the above formulas would result in a stationary point at t = 0, which 
leads to the incorrect approximation d m e i x / 4 . For this integral, there 
are stationary Points at both t = 0 und t = X. Using the contributions from 
both of these stationary Points results in the approximation 
[2] If there are no stationary Points in the interval of integration, then the 
leading Order asymptotic behavior is determined by t he contribution near 
the limits of integration. The leading Order behavior, in this case, can be 
determined by integration by Parts. 
If the leading Order expansion of f ( t ) , near the critical point t = c, is given [3] 
bY f(t) = f(c) + 7 f'"'(c) (t - c)" + . . ., then the leading Order behavior of I 
is given by (assuming, again, that g(c) # 0): 
Equation (50.2) is just this formula evaluated at n = 2. Note that if the 
stationary point is a boundary point, then the factor of 2 in (50.4) does not 
appear . 
As an example, consider the Bessel function at large Order and large ar- 
gument. That is, consider the integral (See (50.3)) J,(rn) = 7r-' JOT cos(rnt- 
m sint) dt , for rn > 1. Writing this as Jm(rn) = (27r)-l E, JOT e*im(t-sint) 
dt , we identify f ( t ) = t - sint. For this integral, the only stationary point 
is at c = 0 (since f'(c) = 0). At this stationary point the second derivative 
vanishes: f"(0) = 0. Directly keeping terms of the next Order results in 
This result could also have been obtained from using (50.4). We have 
g ( t ) = 1/7r, f ( t ) = sint - t , f'(t) = cost - 1, f " ( t ) = -sint, and f"'(t) = 
- cost. Hence, we find that c = 0, n = 3, and If"'(c)I = 1. Using (50.4) and 
50. Stationary Phase 229 
removing the factor of 2 since the stationary point is a limit of the integral 
we find 
1 r (L) 3 113 - -3 (;) COS (-;) 
n 3 
[4] Stationary phase is an application of the method of steepest descents, See 
page 229. In the method of steepest descents, the function f(z) can be 
complex valued. In Laplace’s method (page 221), the argument to the 
exponential is purely real. 
This method was first applied in Stokes [9]. A rigorous justification of the 
method is presented in Watson [ l O ] . 
[5] 
References 
[l] 
[2] 
[3] 
[4] 
M.Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 
National Bureau of Standards, Washington, DC, 1964. 
C. M. Bender and S. A. Orszag, Advanced Mathematical Methods f o r Scien- 
tists und Engineers, McGraw-Hill, New York, 1978. 
N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, 
Dover Publications, Inc., New York, 1986, Chapter 6, pages 219-251. 
R. A. Handelsman and N. Bleistein, “Asymptotic Expansions of Integral 
Transforms with Oscillatory Kernels; A Generalization of the Method of 
Stationary Phase,” S I A M J. Math. Anal., 4, No. 3, 1973, pages 519-535. 
N. Chako, “Asymptotic Expansions of Double and Multiple Integrals Oc- 
curring in Diffraction Theory,” J. h s t . Maths. Applics, 1, December 1965, 
pages 372-422. 
J. C. Cooke, “Stationary Phase in Two Dimensions,” I M A J. Appl. Math., 
29, 1982, pages 25-37. 
F. De Kok, “On the Method of Stationary Phase for Multiple Integrals,” 
SIAM J . Math. Anal., 2, No. 1, February 1971, pages 76-104. 
J. P. McClure and R. Wong, “Two-Dimensional Stationary Phase Approx- 
imation: Stationary Point at a Corner,’’ S I A M J. Math. Anal., 22, No. 2, 
March 1991, pages 500-523. 
[9] G. G. Stokes, “On the Numerical Calculation of a Class of Definite Integrals 
and Infinite Series,” Camb. Philos. Trans., 9, 1856, pages 166-187. 
[ l O ] G. N. Watson, “The Limits of Applicability of the Principle of Stationary 
Phase,” Proc. Camb. Philos. SOC., 19, 1918, pages 49-55. 
[5] 
[6] 
[7] 
[8] 
230 IV Approximate Analytical Methods 
51. Steepest Descent 
Applicable to Integrals of the form l c e x f ( z ) g ( x ) d x , as X tends to 
infinity, where C is a contour in the complex plane and f(x) and g(x ) may 
be complex. 
Yields 
An asymptotic approximation when X > 1. 
Idea 
Given an integral in the form of lc ex f ( ' )g(z ) dz deform the contour 
of integration so that it is in the form of a Laplace integral (See Page 221) 
and apply the method described there. 
Procedure 
Some definitions are needed before the method can be described. Given 
the complex analytic function f(z), of the complex variable z (i.e., z = 
x + iy), let fR and fr denote the real and imaginary parts of that function 
(i.e., f(z) = f ~ ( z ) + ifr(z)). Given a point XO, a directed curve from zo 
along which f ~ ( z ) is decreasing is called a path of descent. The path of 
steepest descent is the curve whose tangent is given by -vfR. This path 
is also one of the curves along which fl(z) is constant. That is, a curve of 
steepest descent is also a curve of constant phase. 
Suppose that f ( z ) and its first n - 1 derivatives vanish at z = zo: 
= 0, for q = 1 , 2 , . . . , n - 1, 
(51.1) 
That is, f (z) = f(z0) + f("'(xo)(z - zo)"/n! +. . . . If z = 20 +Eeie, then the 
directions of constant phase (i.e., fr is constant), from the point z = zo, 
are given by 
(51.2) 
for p = O , l , . . . , n - 1. Furthermore, fR decreases where cos(noP + a ) 
is positive, and increases where cos(nOP + a) is negative. Using a = 0 
for illustrative purposes, Figure 51.1 shows the regions of increasing and 
decreasing values of fR. (A nonzero value for a would just rotate the 
shaded regions in Figure 51.1.) 
The method of steepest descents can be succinctly described by the 
following steps: 
51. Steepest Descent 231 
. . . . . . . . . 
Figure 51.1 For a = 0, the regions of increasing fR are shown clear, the regions 
of decreasing fR are shown shaded. The paths of steepest descent are shown 
dashed, the paths of steepest ascent are shown dotted for (a) n = 2 and (b) 
n = 3. 
Identify the possible critical Points of the integrand. These are the 
endpoints of integration, Singular Points of f (z) and g(z ) and saddle 
Points of f(z). (A saddle Point is a Point at which d f / d z vanishes.) 
Determine the paths of steepest descent from each of the critical Points. 
A picture of the complex plane with curves drawn is usually very 
helpful. 
Justify, via Cauchy’s theorem, the deformation of the original contour 
of integration C onto one or more of the steepest descent paths that 
were determined that connect the endpoints of integration. 
The original integral is then equal to any contributions determined 
from the application of Cauchy’s theorem and an integral along every 
steepest descent contour that has been used. Each of these integrals 
has the appropriate form so that Laplace’s method may be used to 
determine the asymptotic value (see Page 221). 
Special Case 
In the Special case that n = 2, then we can formally proceed as follows. 
The function f(z) can be expanded about the Point z = zo to find f (z ) - 
f(z0) = +f”(zo)(z - ~ 0 ) ~ + . . .. If Q! = argf”(z0) and z = zo + &eie, then 
this expansion can be written 
ia 2 2ie f(z) - f(z0) = $ If”(zo)l e E e + . . . 
= $ If”(z0)l (cos(a + 2e) + isin(a! + 20)) + . . . . 
The steepest paths are given by sin(a + 20) = 0. The paths of steepest 
descent are given by cos(a + 20) < 0. For n = 2, the complex plane has 
the appearance shown in Figure 51.1.a. If the contour can be made to pass 
through the saddle Point, via curves of steepest descent, then the integral 
232 IV Approximate Analytical Methods 
will have the form 
I ( X ) = g ( z ) e x f ( z ) d z 
g ( x ) e x f ( z ) dz. 
J, 
g ( z ) e x f ( ’ ) dz - 
= L a t h into saddle L a t h out of saddle 
Let A represent a point from which the contour enters the saddle, and 
let B represent a point that the contour approaches after leaving the 
saddle. Then the last integral can be written as I (A) = f g ( z ) e ’ f ( ’ ) dz - 
Jt g ( z ) e x f ( ’ ) dz. Changing variables by -7 = f ( x ) - f (zo), these integrals 
can be written as 
dz 
d r 
(51.3) 
The Laplace type integrals in (51.3) will be dominated by the contributions 
near r = 0, which corresponds to z = Z O . Near this point we have r = 
-(f(z) - f(z0)) = -i f ” ( z o ) ( z - ~ 0 ) ~ + . . .. Inverting this relation we find 
LA dz dT A I ( X ) = e x f ( z o ) Jo e - x T g ( z ( 7 ) ) - drr - e x f ( Z 0 ) e - ’Tg(x(7>) - d7. 
This can be used to evaluate the dz/dr in (51.3). Substituting for this term, 
and extending both A and B to 00 (to get the leading Order approximation) 
results in 
(51.4) 
This result should be used with caution because of the many assumptions 
in its derivation. It is usually best to work out each Problem from scratch, 
without resorting to formulas like (51.4). 
51. Steepest Descent 233 
Im z : f .. 
i / 
i / 
\ 
\ 
\ 
\ 
\ 
\ 
. \ \ 
C Re z 
0 
- z0 
Figure 51.2 The geometry of the complex plane for the integral in (51.5). 
Example 1 
Consider the integral 
as X -, 00. The integration contour, C, consists of the positive real axis. 
Identifying f(z) = z + iz - z3, we find that the saddle Points are given 
by f‘ = 0 or z = f z o = d 1 2 ~ / ~ 3 - ~ / ~ e ~ ” / ~ . Near the point z = zo we can 
expand f ( x ) - f(z0) = Sf”(zo)(z - Z O ) ~ + . . .. Using z = zo + &eie, this 
expansion can be written as 
f (z ) - f(z0) = -21/431/2~2 
A path of constant phase has Im(f(z) - f(z0)) = 0, or sin(x/8 + 28) = 0. 
A path of descent (as opposed to a path of ascent) has Re(f(z) - f(z0)) 
decreasing, therefore COS (7r/8 + 28) > 0. 
Figure 51.2 Shows the original contour of integration (C), the two 
saddle Points (&zo), and the structure of the saddle around the point 
z = XO. The steepest descent contours CO and C, are also shown in 
Figure 51.2. The immediate Problem is that the given integration contour C 
is not a steepest descent contour. However, using Cauchy’s theorem, we can 
deform the integration contour as long as we account for any singularities 
that we may Cross. Because there are no singularities of the integrand in 
the region bounded by the contours, Cauchy’s theorem tells us that 
234 IV Approximate Analytical Methods 
Therefore, if we can determine the integrals along the steepest decent 
contours then we can determine I ( X ) . 
Changing variables by -r = f (z )- f(zo), we find 
(51.6) 
At this Point we have reduced the Problem to calculation of Laplace type 
integrals. Evaluating these integrals is straightforward. 
Near z = ZO, which is where the dominant contribution of the integrals 
in (51.6) Comes from, we have r = - ~ f ” ( z o ) ( z - Z O ) ~ + . . .. This relation 
can be inverted to find 
where ,ß = 2-1/83-1/4e-i“/16. Therefore, (51.6) becomes 
Example 2 
Consider the integral representation of the Airy function 
J ( s ) = Ai(s) = - 7r /: cos (f + s t ) d t = J, exp (sw - :) dw 
as s tends to positive infinity. The contour of integration is shown in 
Figure 51.3 (See Page 3). Since this integral is not in the form we require, 
we Change variables by z = w& and X = s3l2 to obtain 
(51.7) 
51. Steepest Descent 235 
Figure 51.3 Contour of integration for (51.7). 
. . . . . 
c 4 c3 
6. .->. . . . . . . . . . .+ .......................................... 
Figure 51.4 The geometry of the complex plane for the integral in (51.7). 
Note that the contour remains unchanged. We can now proceed with the 
steps described above. 
In this case, f (z) = z - z 3 / 3 m d g(z ) = 1. Since both f(z) and 
g(z ) are analytic, the only critical Points are the saddle Points of f (z ) and, 
possibly, the endpoints of the contour of integration. The saddle Points of 
f (z ) are given by f'(z0) = 1 - z i = 0, or xo = f l . 
Expanding f(z) into real and imaginary components results in 
f (z ) = z - ;z3 = f&) + if&) = x (1 - 6x2 + y2) - iy (x2 - I 3 9 1)- - 
The curves of steepest ascent and descent at the Points zo = f l , satisfy 
That is, y (x2 - $y2 - 1) = 0. These curves are shown in Figure 51.4. 
We can now use (51.1) and (51.2) to determine exactly what the steepest 
descent paths are from the saddle Points. Since f"(z) = -22 we find that 
f"(z = -1) = -2 = 2eZ" and f"(z = 1) = 2 = 2ei0. Hence, a,=1 = 7r and 
(u,=-~ = 0. We conclude, therefore, that: 
236 IV Approximate Analytical Methods 
/ ................. ................ ............. ...... ..... ..:.:.:$$g::.. 
. . ...... 
....................... .Y;. ............................ 
.+I+. 
....;. 
\ 
/ Re t ..:: $flf j;::.. ............ ..................... ....................... 
A 
Figure 51.5 The geometry of the complex plane for the integral in (51.5). 
[l] The directions of steepest descent from x = -1 lie along the hyperbolas 
2 2 - i y 2 = 1 (these are labeled CI and C2 in Figure 51.4). 
[2] The directions of steepest descent from z = 1 lie along the line y = 0 
(these are labeled C3 and C4 in Figure 51.4). 
Now that we have identified the curves of steepest descent, we would 
like to move OUT given contour to QW or more of these steepest descent 
contours (justified, of Course, by Cauchy’s theorem). Since the integrand 
is analytic (it has no poles or branch Cuts) the original contour C can be 
deformed into the new contour to obtain C = -CI - C4 + C3 (the plus 
and minus signs are needed since the contours shown in Figure 51.4 are 
oriented). We conclude that 
These integrals are easily evaluated by Laplace’s method. We find (See 
Bleistein and Handelsman [3], page 267 for details) 
5 4 / 4 
2 f i 
This leads to Ai(s) N -e-(2s2’3/3) as s + 00. 
Example 3 
Consider the integral for the reciprocal of the Gamma function (See 
(51.8) 
where X may be real or complex, the principal value of the logarithm is 
taken, and the contour C is shown in Figure 51.5. (The contour C Starts at 
-00, circles the origin once in the positive sense, and returns to -00.) We 
are interested in (51.8) as X + 00. Changing variables by t = Xx results in 
t he representation 
51. Steepest Descent 237 
F’rom this expression we have g(z ) = and f (2) = x - log z. The only 
stationary Point is at f’(z0) = 0 or zo = 0. At the stationary Point we 
find f”(z0) = 1. A simple computation Shows that the paths of steepest 
descent are as indicated in Figure 51.5. 
The contour C in Figure 51.5 can be trivially moved to go through the 
stationary Point. Hence, the asymptotic behavior of I ( X ) is determined 
by the two paths of steepest descent indicated in Figure 51.5. Using the 
Substitution z = 1 + is, and then the expansion log(1 + x) = x - i x 2 + 
i x 3 - . . ., we find 
If the above Laplace type integrals were expanded to higher Order, 
then we would obtain the expansion 
Example 4 
For this 
integrand, the half-plane with Imz > 0 is a region of descent. The steepest 
descent contours are the lines with Rex equal to a constant. In this case 
the integration contour C is deformed into the 3 contours Ci , Ca, and C3 (see 
Figure 51.6). By Cauchy’s theorem we have I ( X ) = sc = sc, + sc2 + Jc3, 
where the integrand is the Same for each integral. In the limit of the 
rectangle extending vertically to infinity, we find that I (A) = R” + sl+im. 
The first of these integrals can be evaluated exactly 
Consider the integral I ( X ) = s: log zeiXZ dz, for A > 1. 
1 
(51.10) 
l0gX i r + n / 2 - - -i- - 
x x 
238 IV Approximate Analytical Methods 
I. 
Re z 
Figure 51.6 Integration contour and steepest descent contours for s: log zeiXZ dz. 
where 7 is Euler's constant. 
Using log(1 + is) = - CF=l(_is)n/n, we find 
The second integral in the above can be evaluated by Watson's lemma. 
log zeiXZ dx = -i log(1 + i ~ ) e z ~ ( ~ + Z ~ ) ds 
(-i)"(n - l)! 
l+i00 
00 
N ieiX c asX+oo. 
P 
n=l 
(51.11) 
Combining the results in (51.10) and (51.11) results in an asymptotic 
expansion of I ( X ) . Full details may be found in Bender and Orszag [l], 
pages 281-282. 
Program 51 
s: if mod(capN,2)=0 
then 1 
else ( if tpos then 1 else -1) $ 
wp [k] : = block ( [templ , 
if k=0 
then w (XI 
else (temp:wp [k-11 , ratsimp(dif f (temp ,x) $ 
51. Steepest Descent 239 
term(m,k,capN,kp):=((m-k-kp)*(m+l)/capN+kp) * 
(m-kp) ! /k!*alphaCm-k-kpl *DCm,m-kpI $ 
D [m , k] : = block ( [kp] , 
if m=k 
then gamma( (m+l)/capN )/(capN*m!) 
else factor( -l/(m-k>*sum( term(m,k,capN,kp) , kp,O,m-k-l) I)$ 
hold : -s*wpz CcapNl *name/capN ! $ 
specialsum3 (mp) :=sum( D C2*mp, kl *phipz Ckl , k, 0 2*mp) $ 
Xe** (name*w (20) * sum( 2* hold** (- (2*mp+l) /capN) * specialsum3 (mp) , 
mp , 0 ,numberof terms) ; 
Notes 
This is also called the saddle point method. 
Bleistein and Handelsman [3] treat the more general Problem 
I ( X ) = Jcg( z )H(Aw(z ) )dz where g(z) and w(z) are analytic in the region 
containing the contour C, and X -f 00. Here H ( z ) is an entire transcendental 
function, such as sin z, COS z , or Ai(z). 
In many specific cases, it is often impractical to construct paths of steepest 
descent. In some Problems it may be easier to use Perron’s method (See 
Perron [6]), which avoids the explicit construction of steepest descent paths. 
See Wong [7] for details. 
Campbell, F’röman, and Walles [4] give explicit series for the asymptotic 
approximation of J e-’”(”)4(z) dz as X -f 00 for a well isolated Singular 
point 20. A Computer program written in REDUCE is also given. 
If w(j)(zo) = 0 for j = 1,2,. . . , N - 1 and w ( ~ ) ( z o ) # 0 then the only 
inputs required for the program are w(z) and 4 ( z ) (and their derivatives), 
N , 20, the number of terms desired in the expansion, and the sign of the 
Parameter in the reparametrization of w(z) near the Singular point. That 
is, near zo we write w(z) = ~ ( z o ) - stN where 
+1 when N is even, 
-1 
s = { +1 when N is odd and t > 0, 
when N is odd and t < 0. 
The author has re-written the program in Macsyma; it is shown in 
Program 51. As an illustration of the Macsyma program, consider the 
calculation in Example 3 (with X replace by -v). If the following program 
Segment is placed before the Code shown in Program 51: 
capN : 2$ 
20: l$ 
numberofterms:3$ 
tpos : f alse$ 
w(x) :=x-log(x)$ 
phi (x) :=l$ 
name : nu$ 
then the following Output is obtained: 
240 IV Approximate Analytical Methods 
sqrt(- nu) 3/2 5/2 
12 (- nu) 288 (- nu) 
139 sqrt(2) sqrt(%pi) nu 
1 Xe - .....................7/2 
51840 (- nu) 
This result must be multiplied by - to obtain the answer in (51.9). 
27riP-l 
References 
C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scien- 
tists and Engineers, McGraw-Hill, New York, 1978. 
N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, 
Dover Publications, Inc., New York, 1986. 
N. Bleistein and R. A. Handelsman, “A Generalization of the Method of 
Steepest Descent,” J. Inst. Math. Appls., 10, 1972, pages 211-230. 
J. A. Campbell, P. E. F’röman, E. Walles, “Explicit Series Formulae for the 
Evaluation of Integrals by the Method of Steepest Descents,” Stud. Appl. 
Math., 77, 1987, pages 151-172. 
E. T. Copson, Asymptotic Expansions, Cambridge University Press, New 
York, 1965. 
0. Perron, “Über die näherungsweise Berechnung von Funktionen großber 
Zahlen,’’ Sitzungsber. Bayr. Akad. Wissensch., Münch. Ber., 1917, pages 
R. Wong, Asymptotic Approximation of Integrals, Academic Press, New 
York, 1989. 
19 1-2 19. 
52. Approximations: Miscellaneous 
Idea 
niques. 
This sect ion cont ains some miscellaneous integral approximat ion tech- 
Procedure 
then it can be expanded in a Taylor series. Hence, I can be re-written as 
Consider the integral I ( s ) = S , ” e - s t f ( t ) g ( t ) d t . If f ( t ) is analytic, 
(52.1) 
where g(s) = e - s t g ( t ) d t is the Laplace transform of g ( t ) . 
52. Approximations: Miscellaneous 241 
Table 52. Some infinite series expansions for integrals of the form JOm f ( t ) g ( t ) d t . 
Setting s = 0 in (52.1) allows an infinite series expansion for I ( 0 ) = 
JOmf(t)g(t)dt to be obtained. Each choice of g(t) results in a different 
expansion, some are given in Table 52. These results are only formally 
correct; in practice, the resulting expressions may be asymptotically valid. 
Example 
determine that 
Using the fourth expansion in Table 52, with f ( t ) = sint, we readily 
(52.2) 
1 
(2n + i ) m n . 
00 sin mt sin t 
t 
dt = 
n=O 
J := JUm 
(E :>, when m > 1. If this A table of integrals shows that J =T - log - 
result is expanded around m = 00, then the result in (52.2) is obtained. 
1 
2 
Note 
[l] This technique is from Squire [l], who credits Willis [2]. 
References 
[l] 
[2] 
W. Squire, Integration for Engineers und Scientists, American Elsevier Pub- 
lishing Company, New York, 1970, pages 105-107. 
H. F. Willis, "A Formula for Expanding an Integral as a Series," Phil. Mag., 
39, 1948, pages 455-459. 
248 V Numerical Methods: Concepts 
With Peano's kerne1 explicitly determined, we observe that it has a 
constant sign (that is, positive) on the interval [-l,l]. Hence, by carrying 
out the integration in (55.4) we find 
for some E in the range [-1, 13. We could also have obtained the factor of 
$, using (55.5): 
Notes 
[l] The usual way of estimating the error using a quadrature routine is to use 
two different rules, say A and B, and then estimate the error by ( A - BI, or 
some scaling of this. In the case of adaptive quadrature (See page 277), the 
subdivision process provides additional information that can be used. See, 
for example, Espelid and Sorevik [3]. 
Interval analysis is a technique in which an interval which contains the 
numerical value of an integral is obtained, See page 218. 
For most automatic quadrature routines, an absolute error and a relative 
error E,. are input. For the integral I = s," f(x) dx, the routine will compute 
a sequence of values {Rnk,Enk}. Here, Rnk is an estimate of I using n k 
values of the integrand, and Enk is the associated error estimate. The 
routine will terminate (and return a value), when the error criteria 
[2] 
[3] 
is achieved. 
Piessens at al. [ l O ] , Section 2.2.4.1, has a Summary of the asymptotic ex- 
pansion of integration errors. Piessens at al. [ l O ] (page 40) also has the 
(pessimistic) error bound: 
[4] 
Let If(k+l)(x)( 5 Mk+i for U 5 x 5 b. Then the absolute 
error of the positive quadrature rule of precision d > k satis- 
fies: 
56. Romberg Integration / Richardson Extrapolation 251 
Table 56.1. Results of using Romberg integration on (56.5). 
3 0.905330 
5 0.964924 0.984789 
9 0.987195 0.994619 0.997895 
17 0.995372 0.998097 0.999257 0.999710 
33 0.998338 0.999327 0.999737 0.999897 0.999960 
65 0.999406 0.999762 0.999907 0.999964 0.999986 0.999994 
i 
2m 
where fi = f(xi) and xi = U + - ( b - U). The error in this approximation 
is given by (see Page 342) 
(56.4) 
where the {Bi} are the Bernoulli numbers and U < q < b. 
Note that only even powers of h appear in the error formula (56.4). 
Because of this, each extrapolation step increases the accuracy by two 
Orders. The extrapolated values of the integral are then given by 
2i 
Ii-i,m-i - (5 ) Ii-1,m-l 
Ii,m = for i = 1 , 2 , . . .. 
1 - (i), 
It should be noted that the values {Il ,m} are identical to the values obtained 
from using the composite Simpson’s rule. 
Example 
Given t he integral 
1 I = i l & d x = l (56.5) 
we might choose to approximate the value by using Romberg integration. 
Table 56.1 has the result of using Romberg integration on this integral. 
Note that the initial data has (at best) about 11 bits of precision (that 
is, - log, 0.000594), yet the fully extrapolated result has more than 17 bits 
of precision (that is, - log, 0.000006). 
256 
Can you f a c t o r i z e the i n t e g r a n d as 
w ( x ) f ( x ) where f i s m o o t h on Ca,bl, 
and w(x)-cos(ux) or s in(wx)? 
V Numerical Methods: Concepts 
- - Use QAWO 
YES 
Can you f a c t o r i t e t h e i n t e g r a n d 
as w(x) f ( x ) whare f is mnooth 
on Ca,bl, and w(x)=l / (x-c)? 
Figure 57.1. Decision tree for finite-range integration from Piessens at al. [9]. 
U s e QAWC or QAWCE I 
YES 
Do you care about c a a p u t c r time, 
and are you w i l l i n g to do some 
a n a l y s i s o f t h e problern? 
t 
U s e QAGS 1 
NO 
NO 
YES 
I ' 
S p l i t t h e i n t e g r a t i o n 
r r 
Are t h e r e d i s c o n t i n u i t i e s or singu- 
l a r i t i e s o f the in tegrand or o f i ts YES 
d e r i v a t i v e w i t h i n t h e i n t e r v a l , and 
do you know where t h e y a r e ? 
NO" 
range a t t h e p o i n t s where 
d i f f i c u l t i e s occur ,and do 
your a n a l y s i s for each sub- 
i n t e r v a l s e p a r a t e l y . You can 
a l s o use QAGP, which i s t o 
be provided wi th the abec ie- 
sae o f t h e p o i n t s involved. 
Has t h e i n t e g r a n d end-point s ingu- 
lar it irr s? Use QAGS 
59. Software Libraries: Excerpts from GAMS 261 
Excerpts from GAMS 
~ ~~ ~~~ 
H2alal: AutomatG 1-D finite interval quadrature (User need 
only specify required accuracy) , integrand available 
via User-defined procedure 
A614 
QlDA 
QlDAX 
QlDB 
QAG 
QAGE 
QAGS 
QAGSE 
QNG 
DCADRE 
QDAG 
QDAGS 
QDNG 
Collected Algorithms of the ACM 
INTHP: a Fortran subroutine for automatic numerical integration 
in H,. The functions may have singularities at one or both 
endpoints of an interval. Each of finite, semi-infinite, and in- 
finite intervals are admitted. (See K. Sikorski, F. Stenger, and 
J. Schwing, ACM TOMS 10 (1984) pp. 152-160.) 
CMLIB Library (QlDA Sublibrary) 
Automatic integration of a user-defined function of one variable. 
Special features include randomization and singularity weaken- 
ing. 
Flexible subroutine for the automatic integration of a user-defined 
function of one variable. Special features include randomization, 
singularity weakening, restarting, specification of an initial mesh 
(optional), and Output of smallest and largest integrand values. 
Automatic integration of a user-defined function of one variable. 
Integrand must be a Fortran FUNCTION but User may select 
name. Special features include randomization and singularity 
weakening. Intermediate in usage difficulty between QlDA and 
QlDAX. 
CMLIB Library (QUADPKS Sublibrary) 
Automatic adaptive integrator, will handlemany non-smooth 
integrands using Gauss-Kronrod formulas. 
Automatic adaptive integrator, can handle most non-smooth 
functions, also provides more information than QAG. 
Automatic adaptive integrator, will handle most non-smooth in- 
tegrands including those with endpoint singularities, uses extrap- 
olat ion. 
Automatic adaptive integrator, can handle integrands with end- 
point singularities, provides more information than QAGS. 
Automatic non-adaptive integrator for smooth functions, using 
Gauss-Kronrod-Patterson formulas. 
IMSL Su bp rogram Li b rary 
Numerical integration of a function using cautious adaptive Rom- 
berg extrapolation. 
IMSL MATH/LIBRA RY Subprogram Library 
Integrate a function using a globally adaptive scheme based on 
Gauss-Kronrod rules. 
Integrate a function (which may have endpoint singularities). 
Integrate a smooth function using a nonadaptive rule. 
262 
QDAGS 
DEFINT 
DOlAHF 
DOlAJF 
DOlARF 
DOlBDF 
QlDA 
ODEQ 
QUAD 
RQUAD 
H2A1 
HLAlU 
SIMP 
V Numerical Methods: Concepts 
IMSL S TA T / L IB R A R Y Su bprogram Li brary 
Integrate a function (which may have endpoint singularities). 
JCAM Software Library 
Uses double exponential transformation of Mori to compute def- 
inite integral automatically to user specified accuracy. 
NAG Subprogram Library 
Computes a definite integral over a finite range to a specified 
relative accuracy using a method described by Patterson. 
1s a general-purpose integrator which calculates an approxima- 
tion to the integral of a function F(x) over a finite interval (A,B). 
Computes definite and indefinite integrals over a finite range to a 
specified relative or absolute accuracy, using a method described 
by Patterson. 
Calculates an approximation to the integral of a function over 
a finite interval (A,B). It is non-adaptive and as such is rec- 
ommended for the integration of smooth functions. These ex- 
clude integrands with singularities, derivative singularities or high 
peaks on (A,B), or which oscillate too strongly on (A,B). 
NMS Subprogram Library 
Automatic integration of a user-defined function of one variable. 
Special features include randomization and singularity weaken- 
ing. 
PORT Subprogram Library 
Finds the integral of a Set of functions over the Same interval 
by using the differential equation solver ODES1. For smooth 
functions. 
Finds the integral of a general user defined EXTERNAL function 
by an adaptive technique to given absolute accuracy. 
Finds the integral of a general User defined EXTERNAL function 
by an adaptive technique. Combined absolute and relative error 
control. 
Scientific Desk PC Subprogram Library 
Automatically evaluates the definite integral of a user defined 
function of one variable. 
Automatically evaluates the definite integral of a user defined 
function of one variable. 
SCR UNCH Subprogram Library 
Calculates an estimate of the definite integral of a user supplied 
function by adaptive quadrature. In BASIC. 
I H2ala2: Nonautomatic 1-D finite interval quadrature, inte- 
1 grand available via User-defined procedure 
59. Software Libraries: Excerpts from GAMS 263 
QK15 
QK21 
QK31 
QK41 
QK51 
QK61 
CMLIB Library (QUADPKS Sublibrary) 
Evaluates integral of given function on an interval with a 15 point 
Gauss-Kronrod formula and returns error estimate. 
Evaluates integral of given function on an interval with a 21 point 
Gauss-Kronrod formula and returns error estimate. 
Evaluates integral of given function on an interval with a 31 point 
Gauss-Kronrod formula and returns error estimate. 
Evaluates integral of given function on an interval with a 41 point 
Gauss-Kronrod formula and returns error estimate. 
Evaluates integral of given function on an interval with a 51 point 
Gauss-Kronrad formula and returns error estimate. 
Evaluates integral of given function on an interval with a 61 point 
Gauss-Kronrod formula and returns error estimate. 
NAG Subprogram Library 
DOlBAF Computes an estimate of the definite integral of a function of 
known analytical form, using a Gaussian quadrature formula with 
a specified number of abscissae. Formulae are provided for a 
finite interval (Gauss-Legendre), a semi-infinite interval (Gauss- 
Laguerre, Gauss-Rational), and an infinite interval 
(Gauss-Hermite). 
NMS Subprogram Library 
QK15 Evaluates integral of given function on an interval with a 15 point 
Gauss-Kronrod formula and returns error estimate. 
H2alb2: Nonautomatic 1-D finite interval quadrature, inte- 
grand available only on a grid 
NAG Subprogram Library 
DOlGAF Integrates a function which is specified numerically at four or 
more Points, over tht: whole of its specified range, using third- 
Order finite-difference formulae wit h error estimates, according 
to a method due to Gill and Miller. 
NMS Subprogram Library 
PCHQA Integrates piecewise cubic from A to B given N-arrays X,F,D. 
Usually used in conjunction with PCHEZ to form cubic, but 
can be used independently, especially if the abscissae are equally 
spaced. 
PORT Subprogram Library 
CSPQU 
H2AlT 
Finds the integral of a function defined by pairs (x,y) of input 
Points. The x’s can be unequally spaced. Uses spline interpola- 
tion. 
Scientific Desk PC Subprogram Library 
Computes the integral of the array f between x(i) and x(j), given 
n Points in the plane (x(k),f(k)), k= l , . . .,n. 
264 V Numerical Methods: Concepts 
H2a2al: Automatic 1-D finite interval quadrature (user need 
only specify required accuracy ) (Special int egrand 
including weight functions, oscillating and Singular 
integrands, principal value integrals, splines, etc.), 
integrand available via user-defined procedure 
CMLIB Library (BSPLINE Sublibrary) 
Integrates function times derivative of B-spline from X1 to X2. 
The B-spline is in B representation. 
Computes integral on (Xl,X2) of product of function and the 
ID-th derivative of B-spline which is in piecewise polynomial 
representation. 
CMLIB Library (QUADPKS Sublibrary) 
BFQAD 
PFQAD 
QAGP 
QAGPE 
QAWC 
QAWCE 
QAWO 
QAWOE 
QAWS 
QAWSE 
QMOMO 
QDAGP 
QDAWC 
QDAWO 
QDAWS 
DOlAKF 
Automatic adaptive integrator, allows user to specify location of 
singularities or difficulties of integrand, uses extrapolation. 
Automatic adaptive integrator for function with user specified 
endpoint singularities, provides more information t hat QAGP. 
Cauchy principal value integrator, using adaptive Clenshaw-Curtis 
method (real Hilbert transform). 
Cauchy principal value integrator, provides more information than 
QAWC (real Hilbert transform). 
Automatic adaptive integrator for integrands with oscillatory sine 
or cosine factor. 
Automatic integrator for integrands with explicit oscillatory sine 
or cosine factor, provides more information than QAWO. 
Automatic integrator for functions with explicit algebraic and/or 
logarithmic endpoint singularities. 
Automatic integrator for integrands with explicit algebraic and/or 
logarithmic endpoint singularities, provides more information than 
QAWS. 
Computes integral of k-th degree Chebyshev polynomial times 
one of a selection of functions with various singularities. 
IMSL MATH/LIBRARY Subprogram Library 
Integrate a function with singularity Points given. 
Integrate a function F(x)/(x-c) in the Cauchy principal value 
sense. 
Integrate a function containing a sine or a cosine. 
Integrate a function with algebraic-logarithmic singularities. 
NAG Subprogram Library 
1s an adaptive integrator, especially suited to oscillating, non- 
Singular integrands, which calculates an approximation to the 
integral of a function F(x) over a finite interval (A,B). 
59. Software Libraries: Excerpts from GAMS 265 
D01 ALF 1s a general purpose integrator which calculates an approximation 
to the integral of a function F(x) over a finite interval (A,B), 
where the integrand may have local Singular behavior at a finite 
number of Points within the integration interval. 
Calculates an approximationto the cosine or the sine transform 
of a function G over (A,B), i.e., the integral of G(x)cos(wx) or 
G(x)sin(wx) over (A,B) (for a user-specified value of U). 
1s an adaptive integrator which calculates an approximation to 
the integral of a function G(x)W(x) over (A,B) where the weight 
function W has end-Point singularities of algebraic-logarithmic 
type (See input Parameter KEY). 
Calculates an approximation to the Hilbert transform of a func- 
tion G(x) over (A,B), i.e., the integral of G(x)/(x-c) over (A,B), 
for user-specified values of A,B,C. 
DOlANF 
DOlAPF 
DOlAQF 
PORT Subprogram Library 
BQUAD Adaptively integrates functions which have discontinuities in their 
derivatives. User can specify these Points. 
~ ~~~ ~~ 
Nonautomatic 1-D finite interval quadrature (spe- 
cial int egrand including weight funct ions, oscillat ing 
and Singular int egrands, principal value integrals, 
splines, etc.), integrand available via user-defined 
procedure 
CMLIB Library (QUADPKS Sublibrary) 
QC25C 
QC25F 
QC25S 
Uses 25 point Clenshaw-Curtis formula to estimate integral of 
F (x) W (x) where W (x) = 1 / (x-c) . 
Clenshaw-Curtis integration rule for function with COS or sin 
factor, also uses Gauss-Kronrod formula. 
Estimates integral of function with algebraic-logarithmic singu- 
larities using 25 point Clenshaw-Curtis formula and gives error 
estimate. 
Evaluates integral of given function times arbitrary weight func- 
tion on interval with 15 point Gauss-Kronrod formula and gives 
error estimate. 
QK15W 
H2a2bl: Automatic 1-D finite interval quadrature (user need 
only specify required accuracy ) (Special int egrand 
including weight functions, oscillat ing and Singular 
integrands, principal value integrals, splines, etc.), 
integrand available only on a grid 
CMLIB Library (BSPLINE Sublibrary) 
BSQAD 
PPQAD 
Computes the integral of a B-spline from X1 to X2. The B-spline 
must be in B representation. 
Computes the integral of a B-spline from X1 to X2. The B-spline 
must be in piecewise polynomial representation. 
266 V Numerical Methods: Concepts 
DCSQDU 
BSITG 
EO2AJF 
EO2BDF 
BSPLI 
SPLNI 
ESHIN 
E3INT 
IMSL Subprogram Labrary 
Cubic spline quadrature. 
IMSL MATH/LIBRARY Subprogram Library 
Evaluate the integral of a spline, given its B-spline representation. 
NAG Subprogram Labrary 
Determines the coefficients in the Chebyshev series representation 
of the indefinite integral of a polynomial given in Chebyshev series 
form. 
Computes the definite integral of a cubic spline from its B-spline 
representation. 
PORT Subprogram Labrary 
Obtains the integrals of basis splines, from the left-most mesh 
point to a specified Set of Points. 
Integrates a function described previously by an expansion in 
terms of B-splines. Several integrations can be performed in one 
call. 
Scaentific Desk PC Subprogram Library 
Evaluates the definite integral of a piecewise cubic Hermite func- 
tion over an arbitrary interval. 
Evaluates the definite integral of a piecewise cubic Hermite func- 
tion over an interval whose endpoints are data Points. 
H2a3al: Automatic 1-D semi-infinite interval quadrat Ure (User 
need only specify required accuracy) (including e-' ) 
weight funct ion) , integrand available via User-defined 
procedure 
~~ ~~ ~ 
Collected Algorithms of the ACM 
A614 INTHP: a Fortran subroutine for automatic numerical integration 
in H,. The functions may have singularities at one or both 
end-Points of an interval. Each of finite, semi-infinite, and in- 
finite intervals are admitted. (See K. Sikorski, F. Stenger, and 
J. Schwing, ACM TOMS 10 (1984) pp. 152-160.) 
OSCINT: a Fortran subprogram for the automatic integration of 
some infinitely oscillating tails. That is, the evaluation of the 
integral from a to infinity of h(x)j(x), where h(x) is ultimately 
positive, and j(x) is either a circular function (e.g., cosine) or a 
first-kind Bessel function of fractional Order. (See J. Lyness and 
G. Hines, ACM TOMS 12 (1986) pp. 24-25.) 
A639 
CMLIB Labrary (QUA DPKS Sublibrary) 
QAGI 
QAGIE 
Automatic adaptive integrator for semi-infinite or infinite inter- 
vals. Uses nonlinear transformation and extrapolation. 
Automat ic int egrator for semi-infinit e or infinite int ervals and 
general integrands, provides more information than QAGI. 
59. Software Libraries: Excerpts from GAMS 267 
QAWF 
QAWFE 
Automatic integrator for Fourier integrals on (a,m) with factors 
sin(wx), cos(wx) by integrating between Zeros. 
Automatic integrator for Fourier integrals, with sin(wx) factor on 
(a,m), provides more information than QAWF. 
IMSL MATH/LIBRARY Subprogram Library 
QDAGI 
QDAWF Compute a Fourier integral. 
Integrate a function over an infinite or semi-infinite interval. 
JCAM Software Library 
DEHINT Uses double exponential transformation of Mori to compute semi- 
infinite range integral automatically to user specified accuracy. 
NAG Subprogram Library 
DOlAMF Calculates an approximation to the integral of a function F(x) 
over an infinite or semi-infinite interval (A,B). 
NMS Subprogram Library 
QAGI Automatic adaptive integrator for semi-infinite or infinite inter- 
vals. Uses nonlinear transformation and extrapolation. 
H2a3a2: Nonautomatic 1-D semi-infinite interval quadrature) 
(including e-2 weight function), integrand available 
via user-defined procedure 
CMLIB Library (QUADPKS Sublibrary) 
QK15I Evaluates integral of given function on semi-infinite or infinite 
interval with a transformed 15 point Gauss-Kronrod formula and 
gives error estimate. 
NA G Su bprogram Li brary 
DOlBAF Computes an estimate G f the definite integral of a function of 
known analytical form, using a Gaussian quadrature formula with 
a specified number of abscissae. Formulae are provided for a 
finite interval (Gauss-Legendre), a semi-infinite interval (Gauss- 
Laguerre, Gauss-rational), and an infinite interval 
(Gauss-Hermite) . 
H2a4al: Automatic 1-D infinite interval quadrature (user need 
only specify required accuracy) (including e-z2 ) 
weight funct ion) , int egrand available via user-defined 
procedure 
Collected Algorithms of dhe ACM 
A614 INTHP: a Fortran subroutine for automatic numerical integration 
in H,. The functions may have singularities at one or both 
end-Points of an interval. Each of finite, semi-infinite, and in- 
finite intervals are admitted. (See K. Sikorski, F. Stenger, and 
J. Schwing, ACM TOMS 10 (1984) pp. 152-160.) 
268 V Numerical Methods: Concepts 
CMLIB Library (QUADPKS Sublibrary) 
QAGI 
QAGIE 
Automatic adaptive integrator for semi-infinite or infinite inter- 
vals. Uses nonlinear transformation and extrapolation. 
Automatic integrator for semi-infinite or infinite intervals and 
general integrands, provides more information than QAGI. 
NAG Subprogram Library 
DOlAMF Calculates an approximation to the integral of a function F(x) 
over an infinite or semi-infinite interval (A,B). 
NMS Subprogram Library 
QAGI Automatic adaptive integrator for semi-infinite or infinite inter- 
vals. Uses nonlinear transformation and extrapolation. 
H2a4a2: Nonautomatic 1-D infinite interval quadrature (in- 
cluding e-22 ) weight function) , integrand available 
via User-defined procedure 
CMLIB Library (QUADPKS Sublibrary) 
QKl5I Evaluates integral of given function on semi-infinite or infinite 
interval with a transformed 15 point Gauss-Kronrod formula and 
gives error estimate. 
DOlBAF 
NAG Subprogram Library 
Computes an estimate of the definite integral of a function of 
known analytical form, using a Gaussian quadrature formula with 
a specified number of abscissae. Formulae are provided for a 
finite interval (Gauss-Legendre) , a semi-infinite interval (Gauss- 
Laguerre, Gauss-rational), and an infinite interval 
(Gauss-Hermite). 
H2blal: Automat ic n-D quadrat Ure (User need only specify 
required accuracy) on one or more hyper-rectangular 
regions, integrand available via User-defined proce- 
dure 
CMLIB Library (ADAPT Sublibrary)ADAPT Computes the definite integral of a User specified function over 
a hyper-rectangular region in 2 through 20 dimensions. User 
specifies tolerance. A restarting feature is useful for continuing a 
computation without wasting previous function values. 
DBLIN 
DMLIN 
IMSL Subprogram Library 
Numerical integration of a function of two variables. 
Numerical integration of a function of several variables over a 
hyper-rectangle (Gaussian method). 
59. Software Libraries: Excerpts from GAMS 269 
IMSL MA TH/L IBR A R Y Subprogram La brary 
QAND 
TWODQ 
Integrate a function on a hyper-rectangle. 
Compute a two-dimensional iterated integral using internal calls 
to a one-dimensional automatic integrator. 
Attempts to evaluate a double integral to a specified absolute 
accuracy by repeated applications of the method described by 
Patterson. 
Computes approximations to the integrals of a vector of similar 
functions, each defined over t he Same multi-dimensional hyper- 
rectangular region. The routine uses an adaptive subdivision 
strategy, and also computes absolute error estimates. 
Attempts to evaluate a multidimensional integral (up to 15 di- 
mensions), with constant and finite limits, to a specified relative 
accuracy, using an adaptive subdivision strategy. 
Returns an approximation to the integral of a function over a 
hyper-rectangular region, using a Monte-Carlo method. An ap- 
proximate relative error estimate is also returned. This routine 
is suitable for low accuracy work. 
NAG Subprogram Labrary 
DOlDAF 
DOlEAF 
DOlFCF 
DOlGBF 
H2bla2: Nonautomatic n-D quadrature on one or more hyper- 
rectangular regions, integrand available via user- 
defined procedure 
NAG Subprogram Library 
DOlFBF Computes an estimate of a multidimensional integral (from 1 
to 20 dimensions), given the analytic form of the integrand and 
suitable Gaussian weights and abscissae. 
Calculates an approximation to a definite integral in up to 30 
dimensions, using the method of Sag and Szekeres. The region of 
integration is an n-sphere, or by built-in transformation via the 
unit n-cube, any product region. 
Calculates an approximation to a definite integral in up to 20 di- 
mensions, using the Korobov-Conroy number t heoretic met hod. 
DOlFDF 
DOlGCF 
H2blb2: Nonautomatic n-D quadrature on one or more hyper- 
rectangular regions, integrand available only on a 
grid 
IMSL Subprogram Library 
DBCQDU Bicubic spline quadrature. 
IMSL MA TH/L IBR A R Y Su bprogram Li brary 
BS2IG 
BS3IG 
Evaluate the integral of a tensor-product spline on a rectangular 
domain, given its tensor-product B-spline representation. 
Evaluate the integral of a tensor-product spline in three dimen- 
sions over a three-dimensional rectangle, given its tensor-product 
B-spline representation. 
270 V Numerical Met hods: Concept s 
H2b2al: Automatic n-D quadrature on a nonrectangular re- 
gion (user need only specify required accuracy), in- 
tegrand available via User-defined procedure 
A584 
A612 
TWODQ 
DOlJAF 
H2B2A 
~~ 
Coliected Algorithrns of the A C M 
CUBTFU: a Fortran subroutine for adaptive cubature over a tri- 
angle. (See D. P. Laurie, ACM TOMS 8 (1982) pp. 210-218.) 
TRIEX: a Fortran subroutine for integration over a triangle. Uses 
an adaptive subdivisional strategy with global acceptance criteria 
and incorporates the epsilon algorithm to Speed convergence. (See 
E. de Doncker and I. Robinson, ACM TOMS 10 (1984) pp. 17- 
22 .) 
CMLIB Library (T W O D Q Subhbrary) 
Automatic (adaptive) integration of a User specified function 
f(x,y) on one or more triangles to a prescribed relative or abso- 
lute accuracy. Two different quadrature formulas are available 
within TWODQ. This enables a user to integrate functions with 
boundary singularities. 
NAG Subprogram Library 
Attempts to evaluate an integral over an n-dimensional sphere 
(n=2, 3, or 4), to a user specified absolute or relative accuracy, 
by means of a modified Sag-Szekeres method. The routine can 
handle singularities on the surface or at the Center of the sphere, 
and returns an error estimate. 
Scientijic Desk Pc Subprogram Library 
Computes the two-dimensional integral of a function f over a 
region consisting of n triangles. 
H2b2a2: Nonautomatic n-D quadrature on a nonrectangular 
region, t he integrand available via User-defined pro- 
cedure 
~~ 
JCAM Software Library 
DTRIA Computes an approximation to the double integral of f(u,v) over a 
triangle in the uv-plane by using an n2 Point, generalized Gauss- 
Legendre product rule of polynomial degree precision 2n-2. From 
“Computation of Double Integrals over a Triangle,” by F. G. 
Lether, Algorithm 007, J. Comp. Appl. Math. 2(1976), pp. 219- 
224. 
N A G Subprogram Library 
DOlPAF Returns a sequence of approximations to the integral of a function 
over a multi-dimensional Simplex, together with an error estimate 
for the last approximation. 
59. Software Libraries: Excerpts from GAMS 271 
Service routines for quadrature (compute weight 
and nodes for quadrature formulas) 
A647 
A655 
A659 
FQRUL 
GQRCF 
GQRUL 
RECCF 
RECQR 
DOlBBF 
DOlBCF 
GAUSQ 
GQOIN 
GQMll 
Collected Algorithms of the ACM 
Fortran subprograms for the generation of sequences of quasiran- 
dom vectors with low discrepancy. Such sequences may be used to 
reduce error bounds for multidimensional integration and global 
optimization. (See B. L. Fox, ACM TOMS 12 (1986) pp. 362- 
376.) 
IQPACK: Fortran routines for the stable evaluation of the weights 
and nodes of interpolatory and Gaussian quadratures with pre- 
scribed simple or multiple knots. (See S. Elhay and J. Kautsky, 
A Fortran implementation of Sobol’s quasirandom sequence gen- 
erator for multivariate quadrature and optimization. (See P. Brat- 
ley and B. L. Fox, ACM TOMS 14 (1988) pp. 88-100.) 
ACM TOMS 13 (1987) pp. 399-415.) 
IMSL MATH/LIBRARY Subprogram Library 
Compute a Fejer quadrature rule with various classical weight 
functions. 
Compute a Gauss, Gauss-Radeau or Gauss-Lobatto quadrature 
rule given the recurrence coefficients for the monic polynomials 
orthogonal with respect to the weight function. 
Compute a Gauss, Gauss-Radeau or Gauss-Lobatto quadrature 
rule with various classical weight functions. 
Compute recurrence coefficients for various monic polynomials. 
Compute recurrence coefficients for monic polynomials given a 
quadrature rule. 
NA G Su bprogram La brary 
Returns the weights and abscissae appropriate to a Gaussian 
quadrature formula with a specified number of abscissae. The 
formulae provided are Gauss-Legendre, Gauss-rational, Gauss- 
Laguerre and Gauss-Hermite. 
Returns the weights (normal or adjusted) and abscissae for a 
Gaussian integrat ion rule with a specified number of abscissae. 
S ix different types of Gauss rule are allowed. 
PORT Subprogram Library 
Finds the abscissae and weights for Gauss quadrature on the 
interval (a,b) for a general weight function with known moments. 
Finds the abscissae and weights for Gauss-Laguerre quadrature 
on the interval (O,+oo) . 
Finds the abscissae and weights for Gauss-Legendre quadrature 
on the interval ( -1 , l ) . 
272 V Numerical Methods: Concepts 
Notes 
[l] 
[2] 
[3] 
In the excerpts section, ACM TOMS Stands for A C M Trans. Math. Soft- 
ware. 
Software is not listed for all the taxonomy classes that have been established. 
The author thanks Dr. Ronald Boisvert of NIST for making part of GAMS 
available electronically. 
References 
R. F. Boisvert, S. E. Howe, and D. K. Kahaner, “GAMS: A Framework for 
the Management of Scientific Software,’’ A C M Trans. Math. Software, 11, 
No. 4, December 1985, pages 313-355. 
R. F. Boisvert, S. E. Howe, D. K. Kahaner, and J . L. Springmann, Guide to 
Available Mathematical Software, NISTIR 90-4237, Center for Computing 
and Applied Mathematics, National Institute of Standards and Technology, 
Gaithersburg, MD, March 1990. 
CMLIB, this is a collection of Code from many sources that NIST has 
combinedinto a Single library. The relevant sublibraries are 
(A) CDRIV and SDRIV, See D. Kahaner, C. Moler, and S. Nash, Numerical 
Methods und Software, Prentice-Hall Inc., Englewood Cliffs, NJ, 1989. 
(B) DEPAC: Code developed by L. Shampine and H. A. Watts. 
(C) FISHPAK: Code developed by P. N. Swartztrauber and R. A. Sweet. 
(D) VHS3: Code developed by R. A. Sweet. 
IMSL Inc., 2500 Park West Tower One, 2500 City West Blvd., Houston, TX 
77042. 
NAG, Numerical Algorithms Group, Inc., 1400 Opus Place, Suite 200, Down- 
ers Grove, IL, 60515. 
NMS, this is an internal name at NIST. The Code is from D. Kahaner, C. 
Moler, and S. Nash, Numerical Methods und Software, Prentice-Hall Inc., 
Englewood Cliffs, NJ, 1989. 
PORT, See P. Fox, et al., The P O R T Mathematical Subroutine Library 
Manual, Bell Laboratories, Murray Hill, NJ, 1977. 
Scientific Desk is distributed by M. McClain, NIST, Bldg 225 Room A151, 
Gaithersburg, MD 20899. 
SCRUNCH, these are old, unsupported Codes in BASIC. The Codes are 
translations of Fortran algorithms from G. Forsythe, M. Malcom, and C. 
Moler, Computer Methods for Mathematical Computations, Prentice-Hall 
Inc., Englewood‘Cliffs, NJ , 1977. 
60. Testing Quadrature Rules 
Applicable to Numerical approximations to integrals. 
Idea 
Many integrals have been used as examples to test quadrature rules. 
60. Testing Quadrature Rules 273 
Procedure 
As new quadrature rules are developed, they are compared to existing 
quadrature rules in terms of accuracy and efficiency. Many authors have in- 
troduced example integrals to indicate t he Performance of t heir algorithms 
and implementations. We tabulate some of those integrals. 
0.1 2 JT (1 - X)2 + 0.01 dxa 0 Lyness [4] uses the test integral I ( X ) = 
0 Piessens et al. [5] uses the test integrals (the numbers correspond to 
their original numbering, numbers 4-6 represent previous integrals wit h 
different Parameters): 
dx = tan-' ((4 - 7r)4"-') + tan-l (7r4"-l) 4-" l1 (x - :)2 + 16-" 
soT COS (2" sin x) dx = 7r JO (2") 
($)"+1 + (,)"+I 1' 1x - i l " dx = 
l + a 
(1 - :)"+l + (:)"+l 1' 13: - :I" dx = 
1 + a 
l 1 1 7r L1- 1 + x + 2-" d x = 4- 
1' log"-l (5) dx = r (a ) 
sin (2"x) dx = 20 sin (2") - 2" COS (2") + 2ae-20 
400 + 4" 
COS (2" 2) dx = T C O S (2"-'> Jo (2"-') 
dx = 
dx = 23"+1 sooo x 2 e - 2 - a x - "-1 
2 (1 - a)7r dx = 1 (1+10x)2 10" sin(7ra) 
274 V Numerical Methods: Concepts 
f5 o-" 
dx = 17) J, (x - 2) ((.*- 1)2 + 4-") 
0 
1) J; Ix - XIa1 dx feature: singularit y 
2) J; f2(x)dx feature: discont inuous 
3) J: e-aa lz -A I dx feature: CO function 
Berntsen et al. [l] uses the test integrals: 
dx 
1oa5 
5, 12$ (x - Xi)2 + 102a5 
feature: one Peak 
feature: four peaks 
6) Ji 2B(x - X) cos(B(x - X)2) dx feature: nonlinear oscillation 
i f x < X 
exp(a2x) otherwise ' and f2(x) = 
10- 
max(X2, (i - x ) ~ ) 
where B = 
0 
1) JI k2 - sin2t 
2, JI d t 
where 0 < k < 1 and X > 0. 
0 
interval analysis integration package: 
Hunter and Smith [3] use the principal-value integrals: 
dt 
='2 cos( COS t ) 
00 -t2 
Corliss and Ra11 [2] have a collection of test Problems that exercise their 
[3.1,3.2] 
sin x dx 
1) J [0,0.11 
2) J:(B sin(Bx) - A sin(Ax)) dz 
8) J1* 0 1 + x 4 
61. Truncating an Infinite Interval 275 
0, x < 0.3 
1, x 2 0.3 ’ where A = [0.0.1] and B = [3.1,3.2], and f(z) = 
Re ferences 
[l] J. Berntsen, T. 0. Espelid, and T. SGrevik, “On the Subdivision Strategy 
in Adaptive Quadrature Algorithms,” J. Comput. Appl. Math., 35, 1991, 
pages 119-132. 
G . F. Corliss and L. B. Rall, “Adaptive, Self-Validating Numerical Quadra- 
ture,” SIAM J. Sci. Stat. Comput., 8 , No. 5, 1987, pages 831-847. 
D. B. Hunter and H. V. Smith, “The Evaluation of Cauchy Principal Value 
Integrals Involving Unknown Poles,” BIT, 29, No. 3, 1989, pages 512-517. 
J. N. Lyness, “When Not to Use an Automatic Quadrature Routine,” SIAM 
Review, 25, No. 1, January 1983, pages 63-87. 
R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, 
Quadpack, Springer-Verlag, New York, 1983, pages 83-84. 
I. Robinson, “A Comparison of Numerical Integration Programs,” J. Com- 
put. Appl. Math., 2, 1979, pages 207-223. 
[2] 
[3] 
[4] 
[5] 
[6] 
61. Truncating an Infinite Interval 
Applicable to Integrals that have an infinite limit of integration. 
Yields 
An approximating integral, with a bound on the error. 
Idea 
easier comput at ion. 
By truncating an infinite integral, a numerical routine may have an 
Procedure 
An infinite integral can always be truncated to a finite interval. Esti- 
mating the error made in the truncation process establishes the usefulness 
of the truncation. 
Example 
Consider the integral I = e-z2 dx. If we truncate the upper 
limit of integration to be, say, (U, then we have 
X cy I N Ja = 1 -e-x2 dx. 
l + x 
VI 
Numerical Methods: 
Techniques 
62. Adaptive Quadrature 
Applicable to 
Yields 
A numerical quadrature scheme. 
Idea 
If a numerical quadrature scheme does not result in a sufficiently 
accurate numerical approximation, then sampling the integrand at more 
nodes should increase t he accuracy. However, the additional sampling only 
needs to be performed in Problem areas (where the error estimates are 
large) . 
Procedure 
Adaptive quadrature is an automatic procedure for increasing the 
accuracy of a numerical approximation to an integral by increasing the 
number of samples of the integrand. Additional samples only need to be 
taken where the quadrature scheme is having numerical difficulties. Hence, 
the Overall scheme is given by the following steps: 
Integrals in any number of dimensions. 
277 
278 VI Numerical Methods: Techniques 
a b 
Figure 62. The geometry of an adaptive integration computation. The func- 
tion f(z) lies in the shaded triangles; the area of these triangles provide error 
estimates. 
Start with a given integral to be iritegrated over a given interval. 
Use a quadrature rule to approximate the integral over the entire 
interval; this is the global approximation. Estimate the error in this 
approximation; this is the global error. Place the interval, and the 
estimated error on that interval, onto a list. 
If the global error is not small enough, then: 
Done. 
Choose an interval from the list of intervals (presumably, choose 
the one with the largest estimated error). 
Subdivide the Chosen interval. 
Approximate the integral over each of the new sub-intervals, and 
estimate the error in these approximations. 
Update the list of intervals and the estimated error on each sub- 
interval. 
Update the global approximation to the integral and the estimate 
of the global error. Go to step [3]. 
Figure 62 Shows an early Stage in an adaptive integration computation. 
adaptive strategy is to subdivide the largest shaded triangle. For the 
figure shown, the left-most region is the next to be sub-divided. 
Rice [12] determines that there are between 1 and 10 million adaptive 
algorithms that are potentially interesting and significantly different from 
one another. This number arises from the following: 
0 There are at least six different processor components. That is, the 
integration rules Chosen for each sub-interval can be the Same, or the 
sub-intervals can have different rules of the Same Order, or many other 
possibilities. 
0 For each choice of how the rules are to be Chosen, there are at least 
five possibilities for each choice of rule. For example, using only three 
nodes per interval, there are the following methods: Simpson’s, 3-point 
62. 
0 
0 
0 
Adaptive Quadrature 279 
Gams, 3-point Tschebyscheff-Gauss, 3-point Tschebyscheff, and Open 
New t On-Cot es. 
There are at least three ways in which to determine a bound on the 
error. 
There are many types of integrands that the routines could identify and 
handle specially. For inst ance, power-law singularit ies or discont inuous 
integrands might be identified. Note, though, that sometimes singu- 
larities at the endpointsof the region of integration can be ignored and 
the computation will still converge (see, for example, Myerson [lo]). 
There are at least six different data structures for maintaining the list 
of intervals used at a given stage in an adaptive algorithm, including 
ordered lists, Stacks, Queues, and “boxes.” For each data structure, 
there are several significant Variations. 
Rice [12] arrives at the pessimistic conclusion that there are potentially 1 
million research Papers to be written, each with a novel algorithm that, for 
some test cases, is Superior to the other 999,999 algorithms. 
Example 
Consider numerically evaluating the integral I = 1; 4x3 dz = 1 using 
adaptive quadrature. We must choose an integration rule and a measure 
of the error. Using the trapezoidal rule in the form 
we find the error estimate E = -- (b - f (2 ) (C) for some 5 E [U, b]. In this 
12 
example f(2)(z) = 24x so we can bound the error by IEI 5 2(b - ~ ) ~ b . An 
adaptive numerical computation can then proceed as follows: 
[l] We start with one interval equal to the entire range of integration: 
interval { P, 11 } 
error est imat es (2) 
integral estimates { 2) 
I N 2 total error 5 2. 
[2] We choose to subdivide the interval with the largest estimated error. 
Subdividing [0, 11 we find: 
intervals {[0,41, 11) 
integral estimates {0.125,1.125) 
error estimates {0.125,0.250) 
I N 1.25 total error 5 0.375. 
[3] Now the largest error is in the interval [i, 11, so we subdivide it to find: 
int ervals {[o, 3, $1, E, 13) 
integral est imates { 0.125, Ö. 2 734, ÖL 71 09) 
error est imates {0.125,0.0234,0.0313) 
I x 1.1093 total error 5 0.1797. 
280 VI Numerical Methods: Techniques 
[4] Now the largest error is in the interval [0, $ 1 , so we subdivide it to find: 
intervals {[o, + I , Li, 3, 3, [:7 111 
integral est imates { 0.0078, Ö.Ö703,0~2734,0.7109} 
error estimates {0.0078,0.0156,0.0234,0.0313} 
I N 1.0624 total error 5 0.0781. 
int ervals {[o, :I, [ 4 , 21, [27 41, [& 111 
integral estimates {0.0078,0.0703,0.2734,0.2729,0.4175} 
error estimates {0.0078,0.0156,0.0234,0.0034,0.0039} 
I N 1.0419 
[5] Now the largest error is in the interval [a 11, so we subdivide it to find: L L 4 i 3 
total error 5 0.0541. 
At this Point we can conclude that I lies in the range [0.99,1.10]. 
Notes 
The subdivision procedure used in xnost adaptive quadrature Codes is a 
simple bisection of the Chosen interval. Berntsen et al. [2] present an al- 
gorithm in which a subdivision strategy results in three non-equally sized 
sub-intervals. 
When an adaptive algorithm is used, the nodes at which the integrand is 
evaluated cannot be determined beforehand. Therefore, adaptive techniques 
are inappropriate for tabulated integrands. 
Most of the routines in Quadpack (See Piessens at al. [ l l ] ) are based on 
adaptive algorithms. 
Corliss and Ra11 [4], who combine interval analysis methods (See page 218) 
with adaptive integration, estimate that the increase in CPU time for modest 
accuracy requests is about a factor of 3-5, and for stringent accuracy re- 
quests the factor is about 3-15. However, with a stringent accuracy request, 
the width of the final interval is only a few units in the last place (ULP). 
References 
J. Berntsen, “Practical Error Estimation in Adaptive Multidimensional Quad- 
rature Routines,” J. Comput. Appl. Math., 25, No. 3, 1989, pages 327-340. 
J. Berntsen, T. 0. Espelid, and T. Sgrevik, “On the Subdivision Strategy 
in Adaptive Quadrature Algorithms,” J. Comput. Appl. Math., 35, 1991, 
pages 119-132. 
J. Berntsen, T. 0. Espelid, and A. Genz, “An Adaptive Algorithm for 
the Approximate Calculation of Multiple Integrals,” A CM Trans. Math. 
Software, 17, No. 4, December 1991, pages 437-451. 
G. F. Corliss and L. B. Rall, “Adaptive Self-Validating Numerical Quadra- 
ture,” SIAM J. Sci. Stat. Comput., 8, No. 5, 1987, pages 831-847. 
M. C. Eiermann, “Automatic, Guaranteed Integration of Analytic Func- 
tions,” BIT, 29, 1989, pages 270-282. 
P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second 
Edition, Academic Press, Orlando, Florida, 1984, 418-434. 
T. 0. Espelid and T. Sorevik, “A Discussion of a New Error Estimate for 
Adaptive Quadrature,” BIT, 29, No. 2, 1989, pages 283-294. 
63. Clenshaw-Curtis Rules 281 
[8] D. K. Kahaner and 0. W. Rechard, “TWODQD an Adaptive Routine for 
Two-Dimensional Integration,’’ J. Comput. Appl. Math., 17 No. 1-2, 1987, 
pages 215-234. 
[9] W. M. McKeeman, “Algorithm 145, Adaptive Numerical Integration by 
Simpson’s Rule,” Comm. ACM, 5 , No. 12, December 1962, page 604. 
[ l O ] G. Myerson, “On Ignoring the Singularity,” SIAM J. Numer. Anal., 28, 
No. 6, December 1991, pages 1803-1807. 
[ll] R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, 
Quadpack, Springer-Verlag, New York, 1983. 
[12] J. R. Rice, “A Metalgorithm for Adaptive Quadrature,” J. ACM, 22, No. 1, 
January 1973, pages 61-82. 
[ 131 H. D. Shapiro, “Increasing Robustness in Global Adaptive Quadrature 
through Interval Selection Heuristics,” ACM Trans. Math. Software, 10, 
No. 2, 1984, pages 117-139. 
63. Clenshaw-Curtis Rules 
Applicable to One-dimensional definite integrals. 
Yields 
A numerical quadrature scheme. 
Idea 
A function can be approximated by a (finite) linear combination of 
basis functions. If the integrated value of the basis functions is known, 
then an approximation of the integral is obtained. 
Procedure 
Let {&(z)} represent a Set of functions for which the integrals sn := 
s,” &(z) dz are known. To approximate the value of I = s,” f(z) dz, we 
approximate the integrand by 
Knowing the {uk} values, we find that I N als1 + u2s2 + . . . + U N S N . 
For the Clenshaw-Curtis rules, we take {U = -1, b = 1) and use the 
Tschebyscheff polynomials +(z) = T’(z) = COS (ncos-’ z). The first few 
such polynomials are Tl(z) = 1, T2(z) = z, and T ~ ( L c ) = 2x2 - 1. In this 
case we can write the {uk} analytically as 
ak = f ( C 0 S 0) COS(ke) d0. (63.1) 
To determine the value of u k numerically, as defined by (63.1), a technique 
may be used that is exact for trigonometric polynomials (See page 322). 
(That is, if f is a polynomial, then the exact value of ak will be returned.) 
282 VI Numerical Methods: Techniques 
Example 
(The exact value is I = 1.) For the Tschebyscheff polynomials we can 
compute 
Consider the numerical evaluation of the integral I = 4/7r J: d m dx. 
1 
s2 = 1 T3(x)dx = 1 (2x2 - l ) d x = --, 3 
1 
2’ 
1 1 
1 
s3 = 1 Tqx) dx = (4x3 - 32) dx = -- 
For our specific integrand, we can evaluate the integral in (63.1) to obtain 
( 0 if k is odd, 
- - 
l6 if IC is even. I r2(1 - k 2 ) 
Therefore, we can approximate I by the series 
1 16 1 1 16 I N 1 . 0 + - * - - - . o + - . - +.... 
2 I r2 3 2 31r2 
The partial sum of this series after 2 terms is I N $ N 0.81. After 4 terms 
we obtain I N 
Note 
[l] 
References 
[l] 
[2] 
N 1.08. 
This method can be modified to account for integrals with weight functions; 
See Piessens at al. [4]. 
C. W. Clenshaw and A. R. Curtis, ‘LA Method for Numerical Integration on 
an Automatic Computer,’’ Numer. Math., 2, 1960, pages 197-205. 
P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second 
Edition, Academic Press, Orlando, Florida, 1984, pages 86-87, 193-196, and 
Sinc Approximation for the Indefinite Integral,” Math. 
of Comp., 41, 1983, pages 559-572. 
R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, 
Quadpack, Springer-Verlag, New York, 1983, Section 2.2.3, pages 28-39. 
446-449. 
[3] R. B. Kearfott, 
[4] 
64. Compound Rules 283 
64. Compound Rules 
Applicable to 
Yields 
Definite integrals in any number of dimensions. 
A numerical quadrature scheme. 
Idea 
If a numerical quadrature scheme is known for integration over a small 
parallelpiped region, then the rule may be repeatedly applied to find a rule 
applicable over a larger parallelpiped region. 
Procedure 
Suppose we seek to approximatean integral numerically over a large 
region. We may subdivide the region into smaller regions (often called 
“Panels”), and then apply a quadrature scheme in each of the smaller 
regions. The resulting quadrature rule, in the larger region, is called a 
compound rule. 
Example 
mensional integral using three nodes via 
Simpson’s rule, in its most elementary form, approximates a one di- 
To apply Simpson’s rule on a large interval, say from c to d, the larger 
region can be subdivided into smaller regions and Simpson’s rule applied 
in each of the smaller regions. 
If the interval [c,d] is subdivided into two equally spaced intervals, 
[c, e] and [e , d] (with e = ( d + c)/2), then 
= [fo 4- 4fi 4- 2f2 4- 4f3 4- f4 , 
3 1 
where fn = f ( c + nh) and h = (e - c)/2 = (d - c)/4. 
(64.2) 
284 VI Numerical Met hods: Techniques 
When manipulating integration rules, as above, it is often easier to 
just use subscripted variables. For example, the above derivation could be 
written as 
d 
I = J L f ( x ) d x 
- h h 
3 - - (fo + 4fl + f 2 ) + - ( f 2 + 4f3 + f4) 3 
h 
3 = - [fO + 4fl 4- 2 f 2 + 4f3 + f4] . 
If the original interval [ c , d ] had been divided into three equally sized 
intervals, instead of two, then the compound rule obtained would have been 
k k k 
2! 3 (fo + 4 f l + f 2 ) + - 3 (f2 +4f3 + f4) + - 3 (f 4 + 4f5 + fs) (64.3) 
where fm = f (c + mk) and k = ( d - c)/6. The extended form of Simpson's 
rule, that is, the rule applied to n equal-width intervals, is: 
where fm = f (c + mk) and k = ( d - c ) / (2n ) . 
Notes 
[l] 
[2] 
A compound rule is sometimes called a composite rule. 
Compound rules also exist for simplices. See, for example, Lyness [3] or De 
Donker [ 21. 
65. Cubic Splines 285 
References 
[l] 
[2] 
P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second 
Edition, Academic Press, Orlando, Florida, 1984, pages 379-384. 
E. De Doncker, “New Euler-Maclaurin Expansions and Their Application 
to Quadrature Over the s-Dimensional Simplex,” Math. of Comp., 33, 1978, 
pages 1003-10 18. 
J. M. Lyness, “Quadrature over a Simplex: Part 1. A Representation for the 
Integrand Function,” and “Quadrature over a Simplex: Part 2. A Represen- 
tation for the Error Functional,” SIAM J. Numer. Anal., 15, 1978, pages 
122-133 and 870-887. 
J. N. Lyness and A. C. Genz, “On Simplex Trapezoidal Rule Families,” 
SIAM J. Numer. Anal., 17, 1980, pages 126-147. 
A. Ralston, “A Family of Quadrature Formulas which Achieve High Accu- 
racy in Composite Rules,” J. ACM, 6, 1959, pages 384-394. 
[3] 
[4] 
[5] 
65. Cubic Splines 
Applicable t o One-dimensional definite int egrals. 
Yields 
A numerical quadrature scheme. 
Idea 
Given data values of a function (not necessarily equally spaced), a 
cubic spline can be fit to those values, and then the integral of the cubic 
spline can be determined. 
Procedure 
Let [U, b] be a finite interval and assume that we have the Points {zi} 
with U 5 2 1 < 22 < - . - < zn+l 5 b and n 2 2. Given the data values 
{f(xi)}, let S(z) be the natural cubic spline which interpolates those data 
values. That is, 
0 S(z) is a cubic polynomial in every interval [xi, 2 i + 4 (say this poly- 
S(z) matches the data values at the nodes Si(xi) = f(zi) for i = 
At the nodes, S and its first and second derivatives are continuous: 
Si-l(xi) = Si(xi), S;-l(xi) = S;(Q), and S,!!-,(q) = S; ’ (q) for i = 
2 , . . . ,n. 
There is no curvature as the ends of the spline: S”(u) = S”(b) = 0 
(this is what makes the spline “natural’)). 
nomial is Si (2)). 
1, - * - 7 n, arid Sn(%+l) = f(%+l). 
(In this definition we have implicitly assumed that U = 2 1 and b = zn+l, 
if this is not the case, then the range for i changes.) See Figure 65. 
286 VI Numerical Met hods: Techniques 
Sn-i S n 
v v 
n n n A n ... s1 s2 
21 x2 23 Xn- i X n X n + i 
Figure 65. Location of the nodes {zi} and the cubic polynomials {Si}. 
We can now approximate the integral of f by the integral of S and 
introduce an error E in doing so: 
b J,” f (x) dx = / S(x) dx + E. (65.1) 
Example 
Consider approximating the integral I = so sin ~ F X dx. We choose to 
use the equally-spaced Points {xi} = ( 0 , $, i, i,l}. Hence, we have the 
data values { (0, 0 ) , (i, e), (i, 1), (i, $), (1, 0 ) ) . 
We choose to represent the cubic on the interval [zi,xi+l] in the form 
Si(.) = ai(x - xi)3 + bi(x - xi)2 + ci(x - xi) + d i . Using this notation the 
first few equations for the unknowns {ai, bi , cg, di I i = 1, . . . ,4} are 
1 
Sl(0) = f (0) = 0 * di = 0 
Si’(0) = 0 * bi = 0 
Si (i) = f (i) = $ 
Si (i) = s2 (i) 
s; (i) = s; (i) 
s2 (t) = s3 (i) =+ ... 
* 
* 
* 
&U1 + hbl + i C l + dl = q 
&U1 + A b 1 + i C l + dl = d2 
&U1 + $bl + c1 = c2 
Si’ (i) = ss’ (i) 
.... * .... 
* $ ~ l + bi = 2b2 
Completing this list of equations, and solving the resulting linear 
System, results in the approximation 
interval [0, i] : -4.8960(z - 0)3 + 3.1340(z - 0) 
interval [a, i]: -2.0288(z - i)3 - 3.6720(z - f ) 2 + 2.2164(z - i) + 0.7071 
interval [$, $1: 
interval [$, 11: 
2.0288(a: - $)3 - 5.1936(z - f ) 2 + 1 
4.8960(z - $)3 - 3.6720(z - $)2 - 2.2164(z - $) + 0.7071. 
(65.2) 
Now we can determine the approximation to the integral: 
h4 h3 h2 
2 4 
n 
~ ( x ) ~ x N - C a i + y C b i + - C c i + h C d i . (65.3) 
i= 1 i=l i=l i=l 
Evaluating (65.3), when the coefficients are given in (65.2), results in the 
approximation I x 0.6362. (Note that the exact value is I = $ N 0.6366.) 
66. Using Derivative Information 287 
Notes 
[l] Among all functions S that are of class C2 [U, b] which interpolate the data 
values, it is the cubic spline approximation that minimizes the “total curva- 
ture”: c = s,” 1s”(z)l2 dz. 
Cubic splines may be numerically computed by the Software in Press et 
al. [5]. 
Use of cubic splines does not result in a conventional quadrature rule. If the 
numerical approximation is written in the form Ja f(z) dz N Ci wif(zi), 
then the weights {wi} depend on all of the function values {f(zi)}. 
G. H. Behforooz and N. Papamichael, “End Conditions for Interpolatory 
Cubic Splines with Unequally Spaced Knots,” J. Comput. Appl. Math., 6, 
1980, pages 59-65. 
P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second 
Edition, Academic Press, Orlando, Florida, 1984, 62-70. 
P. Dierckx, “Algorithm 003. An Algorithm for Smoothing, Differentiation 
and Integration of Experimental Data Using Spline Functions,” J. Comput. 
Appl. Math., 1, 1975, pages 165-184. 
D. Kahaner, C. Moler, and S. Nash, Numerical Methods und Software, Pren- 
tice-Hall Inc., Englewood Cliffs, NJ , 1989. 
W. H. Press, B. P. Flannery, S. Teukolsky, and W. T. Vetterling, Numerical 
Recipes, Cambridge University Press, New York, 1986. 
P. Rabinowitz, “Numerical Integration Based on Approximating Splines,” 
J. Comput. Appl. Math., 33,No. 1, 1990, pages 73-83. 
M. N. E1 Tarazi and S. Sallam, “On Quartic Splines with Application to 
Quadratures,” Computing, 38, 1987, pages 355-361. 
[2] 
[3] 
b 
References 
[l] 
[2] 
[3] 
[4] 
[5] 
[6] 
[7] 
66. Using Derivative Information 
Applicable to Definite integrals. 
Yields 
A numerical quadrature scheme. 
Idea 
A quadrature rule can use the value of the integrand at the nodes, and 
it can also use the value of the derivative of the integrand at the nodes. 
Procedure 
A quadrature rule in the form 
N M [ f(z> E x w j f ( z j ) -k v j f ’ ( z k ) 
j=l k = l 
can often be determined by making the rule exact for the polynomials 
{l,z,z2,. . .}. 
288 VI Numerical Methods: Techniques 
Example 
Consider an integration rule of the form 
Joo f(x> dx = d ( O > + ßf(1) + rfW + Sf’(1). 
Making this rule be exact for f(x) = (1, z, x2, x3}, we obtain the following 
set of simultaneous equations: 
* l = a + ß 
1 
f (4 = 1 
f ($1 = x * 2 = ß + y + S 
f (2) = x2 * i = ß + 2 S 
f(x) = x3 =+ a =ß+3S. 
These equations can be solved to obtain the approximation: 
This rule is knownas the corrected trapezoidal rule. 
Notes 
[l] Some of the common quadrature rules can be improved by including deriva- 
tive terms. The corrected trapezoidal rule is given in the example. The 
corrected midterm rule is 
J/l f(z)dz f ( f > + ii [f’U) - f’(O)] * 
[2] When either the corrected midterm rule or the corrected trapezoidal rule is 
compounded (See page 282), the derivative at the nodes in common cancel, 
so that only the derivatives at the end Points remain. For example, the 
compounded corrected trapezoidal rule is 
h h2 lb f ( 4 = 2 [fo + 2fl + 2f2 + * * * + 2fn-1 + fn] + 12 [f’(a) - f’(b)] +E 
where fi = f(a + ih) and h = ( b - a ) / n . It can be shown that the error is 
bounded by [EI 5 -h4(b - U) max lf‘4’(z)1 (Davis and Rabinowitz [2], 
page 132). 
The trapezoidal rule, corrected by using both f’ and f“ terms, takes the 
form 
1 
720 a S x S b 
[3] 
f(6)(C) and a < C < b (Davis and Rabinowitz [2], page h7 where E = -- 
100800 
133). 
67. Gaussian Quadrature 289 
Re ferences 
[ 11 
[2] 
[3] 
R. A. Cicenia, “Numerical Integration Formulae Involving Derivatives,” J. 
Inst. Muth. Appls., 24, 1979, 347-352. 
P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second 
Edition, Academic Press, Orlando, Florida, 1984, pages 132-134. 
J . D. Lambert and A. R. Mitchell, “The Use of Higher Derivatives in Quadra- 
ture Formulae,” Comput. J., 5, 1962-1963, pages 322-327. 
67. Gaussian Quadrature 
Applicable to One-dimensional definite integrals. 
Yields 
Integration rules on a finite interval using non-uniformly spaced nodes. 
Idea 
A general expression for approximating an integral is proposed. The 
unknown constants in this expression are determined by making the quadra- 
ture rule exact for polynomials of low degree. 
Procedure 
this will minimize the algebra later. We have 
b Given the integral I = Ja g(x) dx we first map it to the interval [-1, 13; 
so that we can now focus on I = J:lf(t)dt. 
approximate integration rule of the form 
Now we search for an 
n 
I N I = azf(ti). (67.1) 
i= 1 
Since this integration rule has 2n unknown constants (the weights {ai} 
and the nodes {t i}) , we can hope to choose these constants so that the 
integration rule is exact for all polynomials of degree less than or equal to 
2n - 1. It turns out this is always possible. 
For example, with n = 2 we have 
If this formula is to be exact for polynomials of degree less than or equal 
to 3, then {al,a2} and {tl,tz} must satisfy the simultaneous algebraic 
equations: 
290 VI Numerical Met hods: Techniques 
Table 67. Values used in Gaussian quadrature formulas. 
~~ ~~ 
Number of Valid for polynomials 
t e r m Nodes { t i } Weights {ai} up to degree 
2 -0.57735027 1 3 
0.57735027 1 
3 -0.77459667 0.55555855 5 
0 0.88888889 
0.77459667 0.55555555 
4 -0.861 13631 0.34785485 7 
-0.33998104 0.65214515 
0.33998104 0.652 14515 
0.86113631 0.34785485 
f ( t ) = t3 L 1 t 3 dt = 0 = alt: + a2t$ 
These equations have the unique Solution (01 = 0 2 = 1, tl = - t 2 = 
l/& x 0.5773). Hence, we have the approximation 
1 
f ( t ) dt N f(-0.5773) + f (0.5773). 11 
To obtain Gaussian quadrature formulas for larger values of n, we 
must find the solutions to a large Set of nonlinear algebraic equations. The 
results of such a calculation are shown in Table 67. 
Example 
Consider the integral J = COS x dx = 1. We represent the 
numerical approximation obtained by using Gaussian quadrature with n 
nodes by J,. Using 2, 3, and 4 nodes we obtain the approximations 
J N J2 = 0.9957702972 
J x J3 = 1.0000365711 
J N J4 = 0.9999998148. 
67. Gaussian Quadrature 291 
Notes 
[l] The values of the {ti} in (67.1) turn out to be roots of the Legendre 
polynomial Pn (x) . These polynomials are defined by the recurrence relation 
(n + l)Pn+l(Z) = (2n + l)xPn(z) - nPn-l(x) 
with the initial conditions: Po(.) = 1 and Pl(x) = x. Then, for example, 
we can calculate P2(x) = $x2 - !j. The roots of P~(z) are at x = &1/& x 
f0.5773. The next Legendre polynomial is P3(2) = 5x3 - $x; its roots are 
at x = 0 and x = &fi x f0.7746. 
The values of the weights {ai} in (67.1) are also functions of Legendre 
polynomials. If xi is the i-th root of Pn(a), then the corresponding weight, 
[2] 
9 
L 
ai, is given by ai = 2 . For example, when n = 2 we find 
(1 - x:) (P&i)) 
When xi = &1/& this results in C Y ~ = 1. 2 ai = 
(1 - x:> (3xi)2 * 
[3] See the section on generalized Gaussian quadrature (page 291) for the anal- 
ogous technique applied to integrals of the form s,” w(z)f(z) dx, when w(x) 
is a positive weighting function. 
Newton-Cotes rules (See page 319) are also interpolatory, but the nodes are 
Chosen to be equidistant from one another. 
Several modifications of the Gaussian principle have been developed, in 
which some of the nodes or weights, or both, are specified in advance. The 
Radeau formulas use one of the endpoints, the Lobatto formulas use both of 
t he endpoints. 
The simplest Radeau formula has the form st f(x) dx x w1 f(z1) + 
w2 f ( O ) , for some unknown (21, w1, w2). Making this quadrature rule exact 
for f(x) = xk (for k = 0,1 ,2) results in the quadrature rule Jol f(x) dx M 
The Tschebyscheff weight function w(z) = (1 - z2)-1/2 is the only weight 
function (up to a linear transformation) for which all the weights in a n-point 
Gauss quadrature formula are equal. See Peherstorfer [5] and the section on 
Tschebyscheff rules (page 331). 
[4] 
[5] 
$ f($) + a f(0). 
[6] 
References 
[l] 
[2] 
[3] 
[4] 
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 
National Bureau of Standards, Washington, DC, 1964, Table 25.4, page 919. 
W. Cheney and D. Kincaid, Numerical Mathematics und Computing, Second 
Edition, Brooks/Cole Pub. Co., Monterey, CA, pages 193-197. 
P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second 
Edition, Academic Press, Orlando, Florida, 1984, Section 2.7, pages 95-132. 
W. Gautschi, E. Tychopoulos, and R. S. Varga, “A Note on the Contour 
Integral Representation of the Remainder Term for a Gauss-Chebyshev 
Quadrature Rule,” SIAM J. Numer. Anal., 27, No. 1, 1990, pages 219-224. 
F. Peherstorfer, “Gauss-Tschebyscheff Quadrature Formulas,” Numer. Math., 
58, 1990, pages 273-286. 
[5] 
292 VI Numerical Methods: Techniques 
68. Gaussian Quadrature: 
Generalized 
Applicable to Integrals that have a positive weighting function. 
Yields 
Integration rules using non-uniformly spaced nodes. 
Idea 
A general expression for approximating an integral is suggested. The 
unknown constants in t his expression are determined by making the quadra- 
ture rule exact for polynomials of low degree. 
Procedure 
Consider the integral 
where w(x) is a specified positive function. To numerically approximate 
I [ f ] for different functions f(x), we choose to use a quadrature rule of the 
form 
n 
(68.1) 
i=l 
where the weights {wi} and the nodes {xi} are to be determined. These 
values are determined by requiring that (68.1) be exact when f(x) is a 
polynomial of low degree. 
In the section on Gaussian quadrature (page 289) the Same formulation 
was used when w(x) = 1 and U and b were finite. In that section it was 
shown that the nodes {xi} and the weights {wi} are related to the roots of 
Legendre polynomials. In the more general case considered here, it can be 
shown that the weights {wi} satisfy a polynomial specified by the weight 
function w(x). 
In particular, if the inner product of two functions f and g is defined 
by (f, g) := J: w(x)f(x)g(x) dx, and j is any positive integer, then there 
exist j polynomials {pk I k = 1,2, . . . , j } with the k-th polynomial being of 
degree k, such that (pkl , p k z ) = 0 when kl # 162. Such a set of polynomials 
can be constructed via 
PO(4 = 1, 
(68.2) 
68. Gaussian Quadrature: Generalized 293 
where p-l(z) = 0 and 
( 0 for i = 0. (68.3) 
We can now state what the values of {wi} and {zi} are. The {zi I 
i = 1 ,2 , . . . , n } are theroots of the polynomial Pn(z) and the {wi I i = 
1,2, . . . , n} are the unique Solution of the (nonsingular) linear system of n 
simult aneous equations 
for k = 1,2, .. . , n - 1. 
i=l 
The nodes and weights appearing in (68.1) may also be found by 
finding the eigenstructure of a specific matrix. If the tri-diagonal matrix 
J n is defined by 
then the eigenvalues of J n will be the nodes in the quadrature formula, 
{zi}. Corresponding to each of these eigenvalues is an associated eigen- 
vector, vi = ( z I ~ ' ) , z I ~ ~ ) , . . . , z I ~ ~ ) ) ~ (so that Jnvi = zivi). The weights in 
the quadrature formula are then given by wi = (U;')) . See Stoer and 
Bulirsch 131 for details. 
The roots, {zi I i = 1 ,2 , . . . , n}, turn out to be real, simple, and to lie 
in the interval (u,b). The weights satisfy wi 2 0 and the relation 
2 
r h n 
holds for all polynomials p ( z ) of degree 2n - 1 or less. For some specific 
weight functions, w(z), and intervals [u,b], the polynomials {Pk I k = 
1 ,2 , . . . , n} turn out to be classical polynomials. For example, we find 
68. Gaussian Quadrature: Generalized 295 
~~ ~~ 
68.3 Lobatto's integration formula 
2 
Approximation: L1 f(z) dz = - n ( n - 1) 1 
i = 2 
Nodes: xi is the (i - 1)st root of Pn-l(z) 
Weights: wi = 
Error term: Rn = - f 
Reference: Abramowitz and Stegun [l], 25.4.32 
2 
n ( n - 1) (P,-i(xi>)2 
n ( n - 1)322n-1[(n - 2)!14 (2n-2) ([) with -1 < [ < 1 
(2n - i)[(2n - 2)!13 
68.4 Weight function xk 
n 
z k f ( z ) da: = E w i f ( z i ) + Rn 
i= 1 
Poiynomials: qn(z> = dk + 2n + I P ~ ~ ' O ) ( I - 22) 
Nodes: zi is the i-th root of gn(z) 
-1 
n!(k + n)! f ( 2 n ) ( 5 ) [ 
(k + 2n)! 1' with 0 < [ < 1 (k + 2n + 1)(2n)! Error term: Rn = 
Reference: Abramowitz and Stegun [l], 25.4.33 
68.5 Weight funct ion di=% 
Approximation: f ( x ) d G d x = wif(zi) + Rn 
Nodes: zi = 1 - t: where [i is the i-th positive root of Pzn+i(s) 
Weights: wi = 2 [ : ~ 1 ~ ~ + ' ) where {w(2n+i)} are the Gaussian weights of Order 
2 n + 1 
Error term: R, = 
Reference: Abramowitz and Stegun [l], 25.4.34 
n 
1' i= 1 
f('")([) with 0 < [ < 1 24n+3[(2n + 1)!14 
(2n)!(4n + 3)[(4n + 2)!]' 
1 
68.6 Weight function - 
JrG 
Approximation: 1' 2 dx = 2 wif(zi) + Rn 
i= 1 
Nodes: zi = 1 - [: where [i is the i-th positive root of Pzn(z) 
Weights: wi = 2wiZn) where {w( '~)} are the Gaussian weights of Order 2n . - 24n+l 
4n + 1 [(4n)!l2 Error term: Rn = -- [(2n)!13 f ( 2 n ) ( [ ) with 0 < [ < 1 
Reference: Abramowitz and Stegun [l], 25.4.36 
296 VI Numerical Methods: Techniques 
1 68.7 Weight function - 
JcT 
(22 - 1)7r Nodes: xi = COS - 2n 
Weights: wi = 
Error term: Rn = 
Reference: Abramowitz and Stegun [l], 25.4.38 
n 
f(2")(E) with -1 < < < 1 7r 
( 2n)!22n-1 
68.8 Weight function d m 
Approximation: J: f ( x ) d C Z d x = 2 wif(xi) + Rn 
Nodes: xi = COS - 
i= 1 
(2 + 1)7r 
n + 1 - . ~ 
7r (2 + 1)7r Weights: wi = - sin2 - 
Error term: Rn = 
n + l 7r n + l 
2n+l f(")((E) with -1 < < < 1 
(2n)!2 
Reference: Abramowitz and Stegun [l], 25.4.40 
2 68.9 Weight function - 
1 - 2 
Approximation: 1- f ( x ) {& dx = wif(xi) + Rn 
k l . . 
Nodes: xi = cos2 
27r Weights: w ' - - 
Error term: Rn = 
Reference: Abramowitz and Stegun [l], 25.4.42 
% - 2 n + i x i 
f('")((E) with 0 < 5 < 1 7r 
(2n)!24n+1 
68.10 Weight function e-% 
Nodes: xi is the 6 th root of Ln(x) (Laguerre polynomial) 
(n!>2 (2n) Error term: Rn = -f 
Reference: Abramowitz and Stegun [l], 25.4.45 
(5) with 0 < 5 < 00 (2n)! 
68. Gaussian Quadrature: Generalized 297 
68.11 Weight function e-”’ 
00 
Approximation: e-“’ f (2) dx = 2 w i f ( x i ) + Rn 
i=l J -00 
Nodes: xi is the i-th root of Hn(2) (Hermite polynomial) 
2*-ln!fi 
n2Hi- l (z i ) 
Weights: wi = 
n ’ f i (2n) Error term: Rn = * f ([) with -00 < [ < 00 
2 (2n). 
Reference: Abramowitz and Stegun [l], 25.4.46 
The technical conditions required on w(2) for the orthogonal polynomials 
to exist are (See Stoer and Bulirsch [3]): 
0 w(x) 2 0 is measurable on the (finite or infinite) interval [u,b]; 
0 All the moments Jab z k w ( x ) dx for k = 0 , 1 , . , . exist and are finite; 
0 If s(x) is a polynomial which is nonnegative on the interval [U, b] , then 
Press and Teukolsky [2] discuss the numerical development of Gaussian 
quadrature rules when the weight function desired is not one of those in 
Table 68. 
Depending on which polynomials are used to obtain the quadrature rule, 
the rules obtained by this technique are called Gauss-Hermite rules, Gauss- 
Jacobi rules, Gauss-Laguerre rules, Gauss-Legendre rules, etc. 
Frequently, Gaussian integration rules of successively higher Order are tried 
when approximating an integral. Unfortunately, it is most often the case 
that information about the function cannot be re-used when using a higher 
Order rule; that is, the roots of the polynomials at each Order do not overlap. 
See the section on Kronrod rules (page 298) for a Solution to this Problem. 
Jab w(z)s(x) d z = 0 implies that s(x) is identically Zero. 
R e ferences 
[l] 
[2] 
M. Abramowitz and I. A. Stegun, Hundbook of Muthematicul Functions, 
National Bureau of Standards, Washington, DC, 1964. 
W. H. Press and S. A. Teukolsky, “Orthogonal Polynomials and Gauss- 
ian Quadrature with Nonclassical Weight Functions,” Comp. in Physics, 
Jul/Aug 1990, pages 423-426. 
J. Stoer and R. Bulirsch, Introduction to Numericul Analysis, translated by 
R. Bartels, W. Gautschi, and C. Witzgall, Springer-Verlag, New York, 1976, 
pages 142-151. 
A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas, Prentice-Hall 
Inc., Englewood Cliffs, NJ, 1966. 
[3] 
[4] 
298 VI Numerical Methods: Techniques 
69. Gaussian Quadrature: 
Kronrod’s Extension 
Applicable to One-dimensional definite integrals. 
Yields 
A sequence of integration rules starting with a Gaussian rule. 
Idea 
When an integral is to be evaluated by different Gaussian integration 
rules, none of the values of the integrand, except possibly for the x = 0 
value, can be re-used. It is possible to devise interpolatory rules that re-use 
all of the nodes in a Gaussian rule. 
Procedure 
Consider the integral I = s,” f(x)dx. Suppose the Gaussian n-point 
rule, Gn, is used to approximate I numerically. Later, it may be of interest 
to approximate I numerically using m nodes (with m > n). If the Gaussian 
m-point rule, Gm, is used, then the only values of f that were obtained 
using Gn that can be re-used is, possibly, the value x = 0. (The value 
x = 0 can be re-used only if both n and m are odd). If f is an expensive 
function to compute, then it would be useful to re-use many values from 
the Gn computation. 
It is possible to start with the nodes from the rule Gn, {xi I i = 
1 , 2 ,..., n}, and add new nodes {yi I i = 1 , 2 ,..., n + 1) so that all 
polynomials of degree 3n + 1 are integrated exactly, if n is even (degree 
3n + 2, if n is odd). The Kronrod rule, which uses the nodes {xi, y j } to 
evaluate polynomials of maximal degree exactly, will be called K2n+l . Note 
that the weights corresponding to the nodes {xi} in K2n+l will not be the 
Same as the weights corresponding to the nodes {xi} in Gn. 
Notes 
As an example, Table 69 shows the rules G7 and K15. 
Piessens et aE. [ll] contains the numerical values of the nodes and weights 
for the following rules: (G7, K15}, {Gio, K21}, (G15, K31}, {Gzo, K41}, 
The technique in this section may be continued; after the n-point Gaussian 
rule (Gn) is used, and the 2n + 1-Point Kronrod rule (K2n+1) is used, 
additional nodes may be added to interpolate higher Order polynomials. 
Nodes and weights for the sequence of rules {G3, K7, PU,, p 3 1 , . . . , p 2 5 5 } are 
given in Patterson [8]. (Here, the rule & is exact for polynomials of degree 
Similarly, the sequence of rules { Glo, K21, P43, P87} is given in Piessens 
et al. [ii]. 
Kronrod extensions also exist for Gaussian rules with the weight function 
(i - z 2 ) p(for -$ 5 p 5 $1. 
(G257 K51}, arid (G30, KSl}. 
(3k + 1)/2.) 
69. Gaussian Quadrature: Kronrod’s Extension 299 
Table 69. Values used in 7-point Gaussian and 15-point Kronrod quadrature 
formulas. The formulas are symmetric, only the positive nodes are shown. (That 
is, if w i f ( t i ) appears, then so does wi f (-ti).) 
Name Nodes Weights 
7-point Gaussian 0.94910 0.12958 
0.74 1 53 0.2 7970 
0.40584 0.38183 
0 0.41795 
15-point Kronrod 0.94910 0.02293 
0.94910 0.06309 
0.86486 0.10479 
0.74153 0.14065 
0.58608 0.16900 
0.40584 0.19036 
0.20778 0.20443 
0 0.20948 
[4] Instead of using a higher Order rule, the same rule can be re-applied with 
a smaller interval size. For even more accuracy, an extrapolation technique 
can be used, See Page 249. 
Favati et al. [3] derive a Set of symmetric, closed, interpolatory quadra- 
ture formulas on the interval [-1, 11 with positive weights and increasing 
precision. These formulas re-use previously computed functional values. 
They obtain a tree of quadrature rules having 74 elements, 27 leaves, and 
a maximum height of 14. That is, the sequence of height 14 is a collection 
of quadrature rules that use (2,3,5,7,9,13,19,27,41,57,85,117,181,249) 
nodes, and each rule re-uses all tshe nodes from the previous rule. 
Favati et al. [3] performed extensive numerical tests using their new 
rules in place of the Gauss-Kronrod rules in the routines QAG and QAGS, in 
the Computer library Quadpack. For one-dimensional and two-dimensional 
integrals, the resulting programs appear to be faster, more reliable, and to 
require fewer function evaluations. 
Rabinowitz [12] considers the numerical evaluation of integrals of the form 
[5] 
[6] 
w(x)f(x) dx with w ( x ) = (1 - x2)p-1/2 for 0 5 p 5 2. f: x - x 
References 
P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second 
Edition, Academic Press, Orlando, Florida, 1984, pages 106-109. 
F. Calib and W. Gautschi, “On Computing Gauss-Kronrod Quadrature 
Formula,” Math. of Comp., 47, No. 176, 1986, pages 639-650, S57-S63. 
P. Favati, G. Lotti, and F. Romani, “Interpolatory Integration Formulas 
for Optimal Composition,” and “Algorithm 691: Improving QUADPACK 
Automatic Integration Routines,” ACM Trans. Math. Software, 17, No. 2, 
June 1991, pages 207-217 and 218-232. 
VI Numerical Methods: Techniques 
W. Gautschi and S. E. Notaris, “An Algebraic Study of Gauss-Kronrod 
Quadrature Formulae for Jacobi Weight Functions,” Math. of Comp., 51, 
No. 183, 1988, pages 231-248. 
A. S. Kronrod, Nodes und Weights of Quadrature Formulas, Consultants 
Bureau, NY, 1965. 
G. Monegato, “Stieltjes Polynomials and Related Quadrature Rules,” SIAM 
Review, 24, 1982, pages 137-158. 
S. E. Notaris, “Gauss-Kronrod Quadrature Formulae for Weight Functions 
of Bernstein-Szego Type. 11,” J. Comput. Appl. Math., 29, No. 2, 1990, 
pages 161-169. 
T. N. L. Patterson, “The Optimal Addition of Points to Quadrature For- 
mulae,” Math. of Comp., 22, 1968, pages 847-856. 
F. Peherstorfer, “Weight Functions Admitting Repeated Positive Kronrod 
Quadrature,” BIT , 30, No. 1, 1990, pages 145-151. 
F. Peherstorfer, “On Stieltjes Polynomials and Gauss-Kronrod quadrature,” 
Math. of Comp., 55, No. 192, 1990, pages 649-664. 
R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, 
Quadpack, Springer-Verlag, New York, 1983. 
P. Rabinowitz, “A Stable Gauss-Kronrod Algorithm for Cauchy Principal- 
Value Integrals.’’ Comput. Math. Appl. Part B, 12, No. 5-6, 1986, pages 
1249-1 254. 
70. Lat t ice Rules 
Applicable to 
unit cube. 
One-dimensional and multidimensional integrals on t he 
Yields 
A numerical approximation scheme. 
Idea 
boundary of the unit cube. 
A lattice rule uses all the nodes on a lattice that lie within and on the 
Procedure 
A lattice rule is a numerical scheme for approximating the value of an 
integral over the multidimensional unit cube. The integral is assumed to 
have the form I = scs f(x)dx, where C” is the closed s-dimensional unit 
cube 
C” = {(XI,.. .,X”) 10 5 xz 5 1, i = 1 ) 2 , . . . , s } . 
Every lattice rule can be written in the form 
70. Lattice Rules 301 
where f is the periodic extension of f (see next Paragraph), { z l , z2 , . . . , Zm} 
are vectors with integer components, and ( 1 2 1 , n2, . . . , nm} are given inte- 
gers called invariants. (It is generally assumed that ni+l divides ni for 
i = 1 , 2 ,..., m - 1 . ) 
Let {x} denote the fractional part of x (e.g., (3.4) = 0.4), and let 
{x) = ({x1,22,. . . ,xs}) = ( { X I } , { x 2 } , . . . , {xs}). Then the periodic ex- 
tension of f is defined by 
- 
f(x) := f({x}), when {xj} # 0, for all j = 1,2, . . . , s. 
Hence, f coincides with f in the interior of the unit cube. At the Points 
on the boundary of the unit cube, f is generally not continuous. At these 
Points, f is defined by 
with each Ti taking only the values f l . If f is continuous on C', then 
this limit exists and is a symmetrical average of the values of f at corre- 
sponding Points on opposite faces of the boundary. For example, in the 
one-dimensional case we find 
Sloan and Lyness [8] consider quadrature rules for the s-dimensional 
hypercube of the form 
which cannot be expressed in an analogous form with a single sum. These 
rules are called rank-2 lattice rules. 
If m summations are required to represent the rule, as shown in (70.1), 
then the rule has rank m. The number of nodes used by such a rule is n:, ni, and the rule may be expressed in a canonical form with m indepen- 
dent summations. Under this classification, an N-node number-theoretic 
rule (see page 312) is a rank m = 1 rule with {n i } = { N , 1, 1, - - - , 1}, and 
the product trapezoidal rule (see Page 323) using N" nodes is a rank m = s 
rule with {ni) = { N , N, - - + , N } . 
302 VI Numerical Met hods: Techniques 
Example 
script has been suppressed) 
In the case of one dimension, (70.1) becomes (where the Single sub- 
l n I N J = - (F) . 3-1 ._ 
Combining this with (70.2) and using z = 1, we are led to the trapezoidal 
rule in the more familiar form 
N-1 
j=l j=l 
The only N-point one-dimensional lattice rule is the trapezoidal rule. 
Notes 
The s-dimensional product-trapezoidal rule is defint 
(70.3) 
+ JA) 
n 
where ej is a unit vector with a one in the xj direction, and Zeros elsewhere. 
For the one-dimensional integration of a periodic function, the trapezoidal 
rule is an efficient choice. However, for s-dimensional integration of a 
periodic function over a hypercube, the s-dimensional product trapezoidal 
rule is not generally cost effective. Other lattice rules can be more effective. 
Lattice rules often have many representations. For example, the two-dimen- 
sional product of the 3-point and 4-point trapezoidal rules may be written 
in the two ways 
12 
(374) 
4 3 & cf ( j 1 7 ("O) + j2v) = ( j 7 ) (70.4) 
Figure 70.a Shows this lattice rule. As another example, all of the following 
represent the Same rule: 
j 1 =l jz=l j=l 
Figure 70.b Shows this lattice rule. 
(70.5) 
70. Lattice Rules 303 
(4 A Y 1 
I 
1 
1 4. . . . . . . . . . . . . . . . . . . . 
I 
I 
I 4. . . . . . ; . . . . . . . . . . . . 4 
2 1 
! I 
I 
A 
Y I 
I 
I : : : : : 
- , . . . .:. . . . . . . . . . . . . . . . . . . . 
1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
4 . . . . . . 
. . . . 
! . . . . . 
2 I , . . . . . . : . . . . : . . . . : . . . . : 
5 : : : : : : . . . . I I . . . . : . . . . . . : . . . * . . . : . . . . : 
. . . . . . . . . . . 
0 L.-:.-.L.,.:,.-.-.L - -> 
0 1 2 3 4 5 5 5 5 1 
Figure 70. The nodes appearing in some lattice rules: (a) the lattice rule in 
(70.4); (b) the lattice rule in (70.5). 
[4] Sloan and Walsh [ l O ] consider lattice rules of rank 2, i.e., rules of the form 
-1. I - 1 (n,l j2z2 -+n), 
Tjz=ljl=l 
where n > 1 and T >_ 1. According to a criterion introduced for number- 
theoretic rules (See page 312) they find the “best rules” of the aboveform. 
Lyness and Smevik [4] use a different measure to determine optimal lattices. 
Both Papers contain numerical results and identify useful rules. 
Lyness and Smevik [3] investigate the number of distinct s-dimensional 
lattice rules that employ precisely N nodes, v s ( N ) . They give many results 
for v, ( N ) including: 
(B) Equality holds in (A) if and only if M and N are relatively prime. 
(C) If the prime factorization of L is L = njp:, then vs(L) = nj vs (p:) 
[5] 
(A) b ( M N ) 5 vs (M)vs (N) , 
s-1 
where vs ( p t ) = n ( Pi+t -‘> - for s 2 1 and t 2 0. 
i=l Pi - 1 
1 For lattice rules of the form 3 f ({ s}), Haber [l] calculates 
several “good” choices for p of the form p = (1, b, b 2 , . ... V-’) for various 
[6] 
values of N and s. Here, goodness is defined by how well the integral rule 
performs on the test function n:=l f(xi), with f(xi) = 1 + n2(x2 - x + $). 
Zinterhof [12] presents a fast method for generating lattices that are nearly 
as good as the optimal lattices. The lattices so generated are tested on the 
integrands 
[7] 
H2(51,. .. ,xs) = n 1 . - + - (1 . 2 
i=l 7 8 2 
304 VI Numerical Methods: Techniques 
@ C”. These integrandsare Chosen because their Fourier transforms are 
Hl(m) = R(m)-’ and H2(m) = R(m)-2, where m = (m1, ..., ms) and 
[SI Sloan and Lyness [9] give some of the properties of the projections of rules 
into lower dimensions. 
[9] Worlet [ll] introduces some new families of integration lattices. These have 
a better Order of convergence that previously known constructions. 
[ l O ] The results in Niederreiter [5] suggest that in the search for efficient lattice 
rules, one should concentrate on lattice rules with large first invariant. 
References 
R(m) = rI;=l max (1, Imzl). 
S. Haber, “Parameters for Integrating Periodic Functions of Several Vari- 
ables,” Math. of Comp., 41, 1983, pages 115-129. 
J. N. Lyness, “An Introduction to Lattice Rules and Their Generator Ma- 
trices,” IMA J . Num. Analysis, 9, No. 3, 1989, pages 405-419. 
J. N. Lyness and T. Smevik, “The Number of Lattice Rules,” BIT, 29, 
No. 3, 1989, pages 527-534. 
J. N. Lyness and T. Sorevik, “A Search Program for Finding Optimal 
Integration Lattices,” Computing, 47, 1991, pages 103-120. 
H. Niederreiter, “The Existence of Efficient Lattice Rules for Multidimen- 
sional Numerical Integration,” Math. of Comp., 58, No. 197, January 1992, 
pages 305-314. 
I. H. Sloan, “Lattice Methods for Multiple Integration,” J. Comput. Appl. 
Math., 12-13, 1985, pages 131-143. 
I. H. Sloan and P. J. Kachoyan, “Lattice Methods for Multiple Integration: 
Theory, Error Analysis and Examples,” SIAM J. Numer. Anal., 24, No. 1, 
1987, pages 116-128. 
I. H. Sloan and J. N. Lyness, “The Representation of Lattice Quadrature 
Rules as Multiple Sums,” Math. of Comp., 52, January 1989, pages 81-94. 
I. H. Sloan and J. N. Lyness, “Lattice Rules: Projection Regularity and 
Unique Representations,” Math. of Comp., 54, No. 190, April 1990, pages 
I. H. Sloan and L. Walsh, “A Computer Search of Rank-2 Lattice Rules for 
Multidimensional Quadrature,” Math. of Comp., 54, No. 189, January 1990, 
pages 281-302. 
R. T. Worlet, “On Integration Lattices,” BIT, 31, 1991, pages 529-539. 
P. Zinterhof, “Gratis Lattice Points for Multidimensional Integration,” Com- 
puting, 38, 1987, pages 347-353. 
649-660. 
71. Monte Carlo Method 
Applicable to 
Yields 
Definite integrals, especially multidimensional integrals. 
A numerical approximation derived from random numbers. 
71. Monte Carlo Method 305 
Idea 
integral. 
Random numbers may be used to approximate the value of a definite 
Procedure 1 
integral 
Suppose we wish to approximate numerically the value of the definite 
(71.1) 
where B is some bounded region. Since B is bounded, it may be enclosed 
in some rectangular Parallelepiped R. Let 1~ (2) represent the indicator 
function of B, that is 
1 i f z E B , 
i n ( x ) = { O i f x # B . 
Then the integral I may be written in the form 
where V(R) represents the volume of the region R. Equation (71.2) may be 
interpreted as the expectation of the function h(X) = g(X)lB(X)V(R) of 
the random variable X, which is uniformly distributed in the Parallelepiped 
R (i.e., it has the density function l/V(R)). 
The expectation of h(X) can be obtained by simulating random devi- 
ates from X, determining h at these Points, and then taking the average of 
the h values. Hence, Simulation of the random variable X will lead to an 
approximate numerical value of the integral I . If N trials are taken, then 
the following estimate is obtained: 
N 
V(R) 5 g(xi)lB(xi) - 1 N I E I = - c h ( X i ) = - 
i= l 
N 
i=l 
(71.3) 
where each xi is uniformly distributed in R. 
Another way to think about (71.3) is that g(&), where & is Chosen 
uniformly in B, is an independent random variable with expectation I . 
Averaging several of these estimates together, which is what (71.3) does, 
results in an unbiased estimator of I . 
306 VI Numerical Methods: Techniques 
Procedure 2 
Importance sampling is the term given to sampling from a non-uniform 
distribution so as to minimize the variance of the estimate for I . Consider 
writing (71.1) as 
I = Eu[g(z)I (71.4) 
where Eu[.] denotes the expectation taken with respect to the uniform 
distribution on B. In other words, I is the mean of g(z) with respect to 
the uniform distribution. Associated with this mean is a variance, defined 
0; := EU [(g(z) - I>'] = Eu [g2] - 12. (71.5) 
bY 
Approximations to I obtained by sampling from the uniform distribution 
will have errors that scale with au. 
If f(x) represents a different density function to sample from, then we 
may write 
where Er[.] denotes the expectation taken with respect to the density f(z). 
In other words, I is the mean of g(z)/f(x) with respect to the distribution 
f(z). Associated with this mean is a variance; defined by 
Approximations to I obtained by sampling from f(x) will have errors that 
scale with a f . 
A minimum variance estimator may be obtained by finding the f(z) 
such that af is minimal. Using the calculus of Variations the density 
function for the minimal estimator is determined to be 
(71.6) 
where the constant C has been Chosen so that fopt(z) is appropriately 
normalized. (Since fopt(z) is a density function, it must integrate to 
unity.) Clearly, finding fopt(x) is as difficult as determining the original 
integral I ! However, (71.6) indicates that fopt(z) should have the same 
general behavior as lg(z)1. As Example 2 shows, sometimes an approximate 
f(z) x fopt(x) can be Chosen. 
71. Monte Carlo Method 307 
Procedure 3 
Another type of Monte Carlo method is the hit-or-miss Monte Carlo 
method (see Hammersley and Handscomb [SI). It is very inefficient but 
is very easy to understand; it was the first application of Monte Carlo 
methods. Suppose that 0 5 f(x) 5 1 when 0 5 x 5 1. If we define 
be estimated by - l n n* 
I = I = - n Cdb-1,Ezi) = - n 
i=l 
(71.7) 
where the {[i} are Chosen independently and uniformly from the interval 
[0,1]. The Summation in (71.7) Counts the number of Points in the unit 
square which are below the curve y = f(z) (this defines n*), and divides 
by the total number of sample Points (i.e., n). We emphasize again that 
the hit-or-miss method is computationally very inefficient . 
Example 1 
To implement the method in (71.3), 
We choose to approximate the integral I = J: 3x2 dz, whose value is 1. 
n 
I = I = - C 3x:, for xi uniformly distributed on [0, 11 
N . a = 1 
the FORTRAN program in Program 71 was constructed. The program 
takes the results of many trials arid averages these values together. Note 
that the program uses a routine called RANDOM, whose source Code is not 
shown, which returns a random value uniformly distributed on the interval 
from Zero to one. 
The result of the program is as follows: 
AFTER 
AFTER 
AFTER 
AFTER 
AFTER 
AFTER 
AFTER 
AFTER 
AFTER 
AFTER 
100 TRIALS, THE AVERAGE IS 1.006200 TRIALS, THE AVERAGE IS 1.084 
300 TRIALS, THE AVERAGE IS 1.046 
400 TRIALS, THE AVERAGE IS 1.033 
500 TRIALS, THE AVERAGE IS 0.996 
600 TRIALS, THE AVERAGE IS 1.028 
700 TRIALS, THE AVERAGE IS 1.035 
800 TRIALS, THE AVERAGE IS 1.029 
900 TRIALS, THE AVERAGE IS 1.032 
1000 TRIALS, THE AVERAGE IS 1.038 
We can also approximate I by using hit-or-miss Monte Carlo. (First, 
we scale the integrand by a factor of 3, to be fs(z) = x 2 , so that it is in the 
range [0, 13.) Now random deviates xi and yi (both obtained uniformly from 
308 VI Numerical Methods: Techniques 
0 X i 
Figure 71. The 323 Points (out of 1000) below the curve y = x2. 
the interval [0, 11) are obtained. For each pair of values, n is incremented by 
one. If, for that pair of values, yi 5 fs(zi) = x:, then n* is also incremented 
by one. Use of (71.7) then results in zn estimate for I . 
Performing this algorithm 1000 times, we obtained 323 instances when 
the yi was less than yz (Figure 71 shows the locations of 
Hence, the estimate of I becomes 
t hese point s) . 
323 = 0.969. 
3 
h N 
h 
Example 2 
Consider the integral 
If we let & represent a sample from the uniform distribution from [ O , l ] 
then J may be approximated by Ju 
N 
J E J u = - ~ C O S - 1 rta 
2 ' 
i=l 
N 
The variance of this estimator for J is 
.0947. . . . o ~ = ~ ' c o s 2 ( ~ ) dx- J 2 = - - - ~ 1 4 
2 7 r 2 - 
Now we Want to obtain a density function that more closely approxi- 
mates the integrand. Since cos (Y) = i - 7 x 2 + 0 ( ~ 4 ) for small values 
of x, we choose a f(z) that has a similar form. We take 
f(x) = C(1- x2) = $(l - z2). 
Ir2 
(71.8) 
71. Monte Carlo Method 309 
1 (The factor arises from the normalization Jo (1 - x2) dx = 9 . ) Using this 
new density function we find (See the Notes for how to generate deviates 
from this distribution) 
(71.9) 
If we let represent a random variable coming from a distribution that has 
the density f(z), then (71.9) may be sampled to yield an approximation 
to J : 
i= 1 
The variance of this second estimator for J is 
2 
2 COS (7) 
crf = 1’ ( - x2 ) f(x) da: - J 2 E .00099.. . . 
Since cru is approximately 10 times larger than af, the errors in using 
J f to approximate J will be about 10 times smaller than using J u to 
approximate J , for the Same number of trials. Of Course, in practical cases 
it will not generally be possible to exactly determine the variances cru and 
c r f . However, estimates can be obtained for the variances by approximating 
t he defining int egrals. 
Program 71 
sm=o 
DO 10 J=l,lOOO 
X=RANDOM (T) 
VAL=3.*X**2 
SüM=SUM+V AL 
IF( MOD(J,100) .NE. 0 GOTO 10 
AVERAG=SüM/FLOAT (J) 
WRITE (6 , 5) J , AVERAG 
5 FORMAT ( ’ AFTER’ ,I6, ’ TRIALS , THE AVERAGE IS ’ , F7.3) 
10 CONTINUE 
END 
310 
Notes 
VI Numerical Methods: Techniques 
This method is of particular importance when multi-dimensional integrals 
are to be approximated numerically. For multi-dimensional integrals, Monte 
Carlo techniques may be the only techniques that will obtain an estimate 
in a reasonable amount of Computer time. This is because the error in a 
Monte Carlo computation scales with a / a , where N is the number of 
samples of the integrand (trials), independent of dimension. For traditional 
methods, the number of samples of the integrand varies exponentially with 
the dimension (i.e., scales as aN for some a). 
While the classical Monte Carlo method converges with Order l/a, where 
N is the number of samples, the quasi-Monte Carlo method can achieve 
an Order of (log N ) ” / N for some a > 0. See Niederreiter [10]-[ll] and 
Wozniakowski’s method on page 333. 
For some integrals, the variance in (71.5) may not exist. For example, with 
I = Ji da:/&, the variance is computed to be 0; = s,’ dz /z - 12, and the 
first term is infinite. Use of importance sampling can result in a new integral 
that has a finite variance. See Kalos and Whitlock [7]. 
Masry and Cambanis [8] discuss how the trapezoidal rule can be used in the 
Monte Carlo computation of the integral I = f(z) da:. Choose n random 
deviates independently and uniformly on the interval [0, 11. Numerically 
Order these deviates to form the sequence tn,l < t n , 2 < . - - < tn,n, and then 
add the Points t„o := 0 and tn,n+l := 1. The sequence of {tn,i}, used in 
the trapezoidal rule, produces an estimate of I: 
If f has a continuous second derivative on the interval [0, 11, then it can be 
shown that 
[fV) - f ’ (0) l2 + 4 1 ) E [I - InI2 = 
4(n + l)(n + 2)(n + 3)(n + 4) * 
Hence, the error varies as 0 (n-4) for large n. 
The integral I = J: g(z) da: may be written as I = J , $ (g(a:) + g(l - a:)) dz. 
Hence, the estimator 
(71.10) 
where the xi are Chosen from the uniform distribution, can be used to 
approximate I. When g(z) is linear, this approximator gives the exact 
answer. (See Siegel and O’Brien [15] for techniques that are exact for other 
polynomials.) In cases where the function is nearly linear, the variance can 
be substantially reduced. This is known as the method of antithetic variates. 
For example, consider the integral I = s,’ e” da: = e - 1. Using a 
straightforward Monte Carlo evaluation the variance is found to be u2 = s,’ [e“ - (e - 1)12 da: = (3 - e ) ( e - 1)/2 = 0.242.. .. Using (71.10) reduces 
the variance to 0.0039, a substantial reduction. 
75. Polynomial Interpolation 319 
[7] P. Keast and J. N. Lyness, “On the Structure of Fully Symrnetric Multi- 
dimensional Quadrature Rules,” SIAM J. Numer. Anal., 16, 1979, pages 
S. L. Sobolev, “Cubature Formulas on the Sphere Invariant Under Finite 
Groups of Rotations,” Soviet Math. DOM, 3, 1962, pages 1307-1310. 
T. SGrevik and T. 0. Espelid, “Fully Symmetric Integration Rules for the 
4-Cube,” BIT, 29, No. 1, 1989, pages 148-153. 
11-29. 
[8] 
[9] 
75. Polynomial Interpolation 
Applicable to Definite integrals. 
Yields 
Integration rules on a finite interval using uniformly spaced nodes. 
Idea 
When values of a function at a discrete set of Points are known, an 
interpolating polynomial can be passed through those Points. The integral 
of the original function will approximate the integral of the interpolating 
polynomial. 
Procedure 
Given the interval [u,b], discretize it into n Segments of equal length 
by inserting the n + 1 nodes: {Xi I xi = U + ih, i = 0, 1 , . . . , n} where 
h = ( b - u) /n . If the function f(x) is known at the n + 1 nodes (i.e., 
fi = f(xi)), then the interpolatory polynomial Pn(x), of degree n or less, 
that goes through all n+ 1 pairs (xi, fi) is given by Lagrange’s interpolation 
(since Li(xj) = Sij, where S i j is the Kronecker delta). Writing x = U + th, 
t-lc 
and using xi = U + ih, this can be written as Li(x) = Ki(t) = n - 
i - k ‘ 
k=O 
Integrating Pn(x) from x = U to x = b leads to 
n r n 
k # i 
(75.1) 
n 
i=O 
320 VI Numerical Methods: Techniques 
Table 75.1. Newton-Cotes rules obtained from polynomial interpolation. 
Wi Error Name n 
~~ 
1 L I 2 2 h3 &f(2) (<) trapezoidal rule 
2 1 4 1 3 3 3 h5 &f(4) (<) Simpson’s rule 
3 3 9 9 3 8 8 8 8 h5 &f(“)(<) Simpson’s 3/8-rule 
h7 &f(6) (<) Milne’s rule* 4 -s464-- 45 45 45 45 45 
5 95 375 250 250 375 95 - h 7 275 
12096 f ‘6’(<) 288 % 288 288 288 288 
h9 & f(’) (<) Weddle’s rule 6 41 216 27 272 27 216 41 140 140 140 140 140 140 840 
where wi = S,”Ki(t)dt. Since the {wi} do not depend on the function 
f(z), they can be pre-computed. For example, for n = 2 we find: 
wo = Jo2 (-) t - 1 (-) t - 2 d t = i d2 (t2 - 3t + 2) d t = - 1 
w1= Jo2 (5) (E) d t = - L2 (t2 - 2t) d t = - 4 0 - 1 0 - 2 3 
1 - 0 1 - 2 3 
1 
3’ 
This gives rise to the integration rule: 
where h = ( b - u)/2 and fk = f(a + Ich). This is known as Simpson’s rule. 
The error in using the integration rules in (75.1) can be shown to be 
given by Jo” Pn(z) d~ - f(z) dz = hpn+lEnf(pn)(<) Jo” 
for some < E (U, b ) where pn and En are functions of n and not of f(z). 
The approximationsgiven in (75.1) are known as the Newton-Cotes 
rules. Some tabulated values of the {wi}, as well as the corresponding error 
term, are presented in Table 75.1. As indicated in that table, some of the 
Newton-Cotes rules also have other names. 
* Also known as Boole’s rule. 
75. Polynomial Interpolation 321 
Table 75.2. Open Newton-Cotes rules obtained from polynomial interpolation. 
n Wi Error Name 
2 2 h3 i f ( 2 ) ( c ) midpoint rule 
3 3 3 h3 f(’) (6) - 
2 2 
5 14 (4) - 
5 95 (4) - 
h Ef (0 
h T44f (t) 
10 5 5 5 10 T40f (6) 
4 8 - 4 8 
5 5 5 5 5 5 5 
6 33 -= 39 -E 7 41 (6) - 
3 3 3 
24 24 24 24 
Example 
Consider the integral I = - cosxdx = 1. Define I , to be 
the result of using the n-node rule from Table 75.1. Then we obtain the 
following approximations to I : 
I N 1 2 = 0.1063722664 
I N 13 = 1.0379687263 
I N I4 = 1.0163528101 
I N 1 5 = 0.9994771775 
I N 16 = 0.9997082294 
I N I7 = 1.0000059300. 
Notes 
[l] Newton-Cotes rules for large values of n are not often used since some of 
the weights {wi} become negative and numerical cancellation occurs in the 
comput at ion. 
In the above we discretized the interval [U, b] into n + 1 nodes that included 
the endpoints a and b. Hence, the above formulas are sometimes called 
closed Newton-Cotes rules. 
If we only consider the interior nodes, {xi I xi = a+ih, i = 1,. . . , n- 1) 
(where, as before, h = ( b - a ) / n ) , and then approximate the given integral 
by the integral of the interpolating polynomial, then we will have derived 
on Open formula. These formulas are sometimes called Open Newton-Cotes 
rules. The first few such formulas are in Table 75.2. 
Instead of just interpolating the value of f(x) at the nodes {xi}, the values 
of f(x) and f’(x) may be used. For example, if values for the derivatives at 
the endpoints of the interval are given, then the approximate formula 
[2] 
[3] 
322 VI Numerical Met hods: Techniques 
may be used (here, h = ( b - U)). It can be shown that the error is given by 
with 
Gaussian quadrature rules (See page 289) are also interpolatory, but the 
nodes are not equidistant from one another. Instead, the node locations are 
Chosen to make the rule have as high a degree as possible. 
Sometimes a quadrature formula is desired t hat integrates trigonometric 
polynomials, not ordinary polynomials, exactly. A trigonometric polynomial 
of degree m is a linear combination of the functions (1, cosx, sinx, cos2 x, 
COS x sin x, sin2 x, . . . , cosm x, cosm-l z sin x, . . . sinm x} . Equivalently, a 
trigonometric polynomial of degree m is a linear combination of the functions 
{ 1, COS x, sin x, COS 22, sin 22, . . . COS wa, sin mx}. The approximation 
E (u ,b ) . See page 287. 
r 2 l r n 
k=l J O 
where h = 27r/n and ß is any real number satisfying 0 5 ß < h, is exact for 
all trigonometric polynomials of degree n - 1 or less (See Mysovskikh [5]). 
Vanden Berghe et al. [7] consider quadrature rules that exactly inte- 
grate ordinary polynomials and trigonometric polynomials. 
Given data values defined on a Set of nodes, one polynomial can be fit to all 
of the data values, as shown above. Alternatively, the region of integration 
may be broken into smaller regions, with an interpolatory polynomial fit to 
the data values in each sub-region. Köhler [3] considers the case when the 
interpolatory polynomial on a sub-region uses data values from outside that 
sub-region. 
There are many interpolatory formulas, other than polynomials, that can be 
used to interpolate data. For example, De Meyer et al. [4] interpolate a Set of 
values using the function f(x) = ekx uixa. This interpolating function 
can then be integrated to obtain, for example, their modified trapezoidal 
rule: 
where 8 = hk and lc is an arbitrary Parameter. This Parameter is Chosen 
in practice, by minimizing the error term. For the above rule, the leading 
h3 [ 4siny:8/2) Order error term has the form E = - 1 - e2 
where q is in the range xo < 7 < xo + h. 
76. Product Rules 
References 
323 
P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second 
Edition, Academic Press, Orlando, Florida, 1984, pages 74-81. 
F. B. Hiderbrand, Introduction to Numerical Analysis, McGraw-Hill Book 
Company, New York, 1974. 
P. Köhler, “On a Generalization of Compound Newton-Cotes Quadrature 
Formulas,” BIT, 31, 1991, pages 540-544. 
H. De Meyer, G. Vanden Berghe, and J. Vanthournout, “Numerical Quadra- 
ture Based on an Exponential Type of Interpolation,’’ Int. J. Comp. Math., 
38, 1991, pages 193-209. 
I. P. Mysovskikh, “On Cubature Formulas that Are Exact for Trigonometric 
Polynomials,” Soviet Math. Dokl., 36, No. 2, 1988, pages 229-232. 
J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, translated by 
R. Bartels, W. Gautschi, and C. Witzgall, Springer-Verlag, New York, 1976, 
pages 118-123. 
G . Vanden Berghe, H. De Meyer, and J. Vanthournout, “On a Class of 
Modified Newton-Cotes Quadrature Formulae Based Upon Mixed-type In- 
terpolation,” J. Comput. Appl. Math., 31, 1991, pages 331-349. 
76. Product Rules 
Applicable to Multidimensional integrals. 
Yields 
A numerical quadrature scheme. 
Idea 
Suppose two numerical quadrature schemes are known, one for T- 
dimensional Euclidean space and one for s-dimensional Euclidean space. 
The “product” of these two rules can be used to formulate a numerical 
quadrature scheme for ( T + s)-dimensional Euclidean space. 
Procedure 
space. Suppose we have the n-node and rn-node quadrature rules 
Let R (S) be a region in r-dimensional (s-dimensional) Euclidean 
J, f(x) dx = 2 Wjf(Xj), 
j=l 
m (76.1 . ~ - b ) 
k=l 0 J S 
Then an (n + m)-node quadrature rule for the region B = R x S, in an 
(r + s)-dimensional space, is given by 
n m 
324 VI Numerical Methods: Techniques 
Example 
an integral by 
On the interval [a,b], Simpson’s rule with three nodes approximates 
(76.3) 
where fn = f ( a + nh) and h = ( b - a)/2. Likewise, on the interval [ c , d ] , 
Simpson’s rule with five nodes approximates an integral by 
(76.4) 
where gm = g(c + mk) and k = (d - c)/4. 
following approximat ion of a two-dimensional integral 
Taking the product of the rules in (76.3) and (76.4) results in the 
where hn,m = h(a + nh, c + mk). 
Notes 
If (76.1.a) exactly integrates f(x), and if (76.1.b) exactly integrates g(y), 
and h(x, Y) = f(x)g(y), then (76.2) will exactly integrate h(x, Y). 
This technique can be used for general Cartesian product regions, not just 
parallelpipeds; for instance, circular cylinders, circular cylindrical shell, and 
triangular prisms. 
Stroud [3] analyzes product rules by use of transformations. Suppose the 
region of integration is S, and the integrals of interest have the weight 
function w(x). If the quadrature rule is to be exact for polynomials, then 
I = JJ. - - s w(x) x:1x;2 . . . xgn dx must be integrated exactly for some Set 
of {ai}. If there exists a transformation of the form x = x(u) that turns I 
into the product I = (s w1(u1)91(u1) du1) . . . (~w,(u,)g,(u,) du,) , and if 
suitable formulas are known for these Single integrals, then one has obtained 
a product rule. 
Using product rules, the number of nodes at which the integrand must be 
evaluated grows exponentially with the dimension of the integration. If 
a one-dimensional quadrature rule that uses 19 nodes is the basis for a 
7-dimensional quadrature rule, then 1g7 = log integrand evaluations are 
required. 
The rules devised by this technique are often not the most efficient in terms 
of number of integrand evaluations. 
S 
77. Recurrence Relations 325 
[6] Often, a more computationally efficient quadrature rule for a multidimen- 
sional integral can be found. For example, Acharya and Mohapatra [l] give 
the two-dimensional quadrature rule for analytic functions: 
f ( ~ , z') dz dz' x hh' - 256f00 + 25 (fil + f13 + f31 + f33) 
(76.5) 
where f a p = f(za,z$), za = zo + hki"-', and Z; = zo + h'ki"-'. When 
k = l/a, the rule in (76.5)