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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 
Example