Singular integral eqs ricardo estrada ram kanwal

Dedicated to Ana
- Ricardo Estrada

Dedicated in loving memory of my parents
- Ram P. Kanwal

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Ricardo Estrada
Ram P. Kanwal

Singular Integral Equations

Springer Science+Business Media, LLC

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Ricardo Estrada
Escuela de Matematica
Universidad de Costa Rica
2060 San Jose
Costa Rica

Ram P. Kanwal
Department of Mathematics
Penn State University
University Park, PA 16802
U.S.A.

Ubrary of Congress Cataloging-in-Publication Data
Estrada, Ricardo, 1956Singular integral equations / Rieardo Estrada, Ram P. Kanwal
p.em.
Includes bibliographical referenees and index.
ISBN 978-1-4612-7123-9
ISBN 978-1-4612-1382-6 (eBook)
DOI 10.1007/978-1-4612-1382-6
1. Integral equations. 2. Kanwal, Ram P. II. Title.
QA431.E73 2000
99-050345
515'.45-de21
CIP

AMS Subjeet Classifieations: 30E25, 4SExx, 4SEOS, 4SE10, 62Exx

Printed on acid-free paper.
©2000 Springer Seience+Business Media New York
Originally published by Birkhl!user Boston in 2000
Softcover reprint ofthe hardcover Ist edition 2000
AII rights reserved. This work may not be translated or copied in whole or in par! without the written
permissionofthepublisher Springer Science+Business Media, LLC,
except for brief excerpts in connection with reviews or scholarly analysis.
Use in connection with any form of information storage and retrieval, electronic adaptation, computer
software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the
former are not especially identified, is not to be taken as a sign that such names, as understood by the
Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
ISBN 978-1-4612-7123-9 SPIN 10685292
Reformatted from authors' disk by TEXniques, Inc., Cambridge, MA.

9 8 765 4 3 2 1

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Contents

Preface

ix

1 Reference Material
1.1 Introduction.........
1.2 Singular Integral Equations .
1.3 Improper Integrals . . . . .
1.3.1 The Gamma function.
1.3.2 The Beta function. . .
1.3.3 Another important improper integral .
1.3.4 A few integral identities . .
1.4 The Lebesgue Integral . . . . . . .
1.5 Cauchy Principal Value for Integrals
1.6 The Hadamard Finite Part . . . . .
1.7 Spaces of Functions and Distributions
1.8 Integral Transform Methods
1.8.1 Fourier transform .
1.8.2 Laplace transform
1.9 Bibliographical Notes . . .

1

1
2
3

2

Abel's and Related Integral Equations

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . ..

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7

11
12
13
15
18
26
30
33
35
38
41

43
43

Contents

Vl

2.2
2.3
2.4
2.5
2.6

Abel's Equation. . . . . . . . . . . . . . . . . . . .
Related Integral Equations . . . . . . . . . . . . . .
The equation (5 - t)fJ g(t)dt = f (5) ,
f3 > -1
Path of Integration in the Complex Plane . . .
The Equation r g(z)dz + k r g(z)dz = f(C)

43
46
48
51
52

2.7
2.8
2.9
2.10

Equations On a C osed Curve . . . . . . . . . .
Examples . . . . . . . . .
Bibliographical Notes.
Problems . . . . . . . . .

57
60
64
64

J;

me

JCor

(z-n"

JC~b (~-z)"

.,

3 Cauchy Type Integral Equations
3.1 Introduction...................
3.2 Cauchy Type Equation of the First Kind . . . .
3.3 An Alternative Approach . . . . . . . . . . . .
3.4 Cauchy Type Equations of the Second Kind . .
3.5 Cauchy Type Equations On a Closed Contour
3.6 Analytic Representation of Functions . . . . .
3.7 Sectionally Analytic Functions (z - a)n-v(z - b)m+v

3.8 Cauchy's Integral Equation on an Open Contour. . .
3.9 Disjoint Contours . . . . . . . .
3.10 Contours That Extend to Infinity
3.11 The Hilbert Kernel ..
3.12 The Hilbert Equation
3.13 Bibliographical Notes.
3.14 Problems . . . . . . .

93
100
103
108
113
116
116

4 Carleman Type Integral Equations
4.1 Introduction..................
4.2 Carleman Type Equation over a Real Interval
4.3 The Riemann-Hilbert Problem . . . . . . . .
4.4 Carleman Type Equations on a Closed Contour
4.5 Non-Normal Problems .. . . . . .
4.6 A Factorization Procedure . . . . . . . . . . .
4.7 An Operational Approach. . . . . . . .
4.8 Solution of a Related Integral Equation. . . . . . . .

125
125
127
134

140
145
150
152
161

4.9

Bibliographical Notes .

. . . . . . . . .

4.10 Problems . . . . . . . . . . . . . . . . . . . . . . .

5 Distributional Solutions of Singular Integral Equations
5.1

Introduction......................

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71
71
72
75
78
82
84
91

168

168

175
175


Contents
5.2 Spaces of Generalized Functions . . . . . . .
5.3 Generalized Solution of the Abel Equation .
5.4 Integral Equations Related to Abel's Equation
5.5 The Fractional Integration Operators <l>a . . . •
5.6 The Cauchy Integral Equation over a Finite Interval
5.7 Analytic Representation of Distributions of E'[ a, b]
5.8 Boundary Problems in A[a, b] . .
5.9 Disjoint Intervals . . . . . . . . . . . . . . . . .

5.9.1

Theprob1em[RjFL =hj . . . . . . . .

5.9.2 The equation A 11l'1 (F) + A21l'2(F) = G .
5.10 Equations Involving Periodic Distributions.
5.11 Bibliographical Notes.
5.12 Problems . . . . . . . . . . . . . . . . .

vii
176
180

187
191
196
205
209
217
221
226
231
242
242

6 Distributional Equations on the Whole Line
6.1 Introduction................
6.2 Preliminaries .. . . . . . . . . . . . .
6.3 The Hilbert Transform of Distributions .
6.4 Analytic Representation. . . . . . . . .
6.5 Asymptotic Estimates . . . . . . . . . .
6.6 Distributional Solutions of Integral Equations
6.7 Non-Normal Equations
6.8 Bibliographical Notes.
6.9 Problems . . . . . . .

251
251
252
254
262
264
270

282
286
287

7 Integral Equations with Logarithmic Kernels
7.1 Introduction.................
7.2 Expansion of the Kernel In Ix - y I . . . . .
7.3 The Equation
In Ix - yl g(y) dy = I(x)
7.4 Two Related Operators . . . . . . . . . . .
7.5 Generalized Solutions of Equations with
Logarithmic Kernels . . . . . . . . . . . . . . . .
7.6 The Operator
(P(x - y) In Ix - yl + Q(x, y)) g(y) dy
7.7 Disjoint Intervals of Integration. . . . . .
7.8 An Equation Over a Semi-Infinite Interval . . . . . . . . .
7.9 The Equation of the Second Kind Over a
Semi-Infinite Interval . . . . . . . . .
7.10 Asymptotic Behavior of Eigenvalues.
7.11 Bibliographical Notes.
7.12 Problems . . . . . . . . . . . . . . .

295
295
296

1:

f:

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298

300
302
308
311
313
314
318

322
323


viii

Contents

8 Wiener-Hopf Integral Equations
8.1 Introduction.................
8.2 The Holomorphic Fourier Transform
8.3 The Mathematical Technique . . . . . . . .
8.4 The Distributional Wiener-Hopf Operators
8.5 Illustrations..........
8.6 Bibliographical Notes .
8.7 Problems . . . . . . . . . . .

339

339
340
345
355
361
369
369

9 Dual and Triple Integral Equations
9.1 Introduction...................
9.2 The Hankel Transform . . . . . . . . . . . .
9.3 Dual Equations with Trigonometric Kernels
9.4 Beltrami's Dual Integral Equations .
9.5 Some Triple Integral Equations.
9.6 Erdelyi-Kober Operators . . . .
9.7 Dual Integral Equations of the
Titchmarsh Type . . . . . . . . . . . . .
9.8 Distributional Solutions of Dual Integral Equations
9.8.1 Fractional integration in H~,v . . . . .
9.8.2 Solution of the distributional problem
9.8.3 Uniqueness..
9.9 Bibliographical Notes .
9.10 Problems . . . . . . .

375
375
377
379
382
384

386
390
391
395
399
402
403
403

References

413

Index

423

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Preface

Many physical problems that are usually solved by differential equation
techniques can be solved more effectively by integral equation methods.
This work focuses exclusively on singular integral equations and on the
distributional solutions of these equations. A large number of beautiful
mathematical concepts are required to find such solutions, which in tum,
can be applied to a wide variety of scientific fields - potential theory, mechanics, fluid dynamics, scattering of acoustic, electromagnetic and earthquake waves, statistics, and population dynamics, to cite just several.
An integral equation is said to be singular if the kernel is singular within
the range of integration, or if one or both limits of integration are infinite.

The singular integral equations that we have studied extensively in this
book are of the following type. In these equations f (x) is a given function
and g(y) is the unknown function.

1. The Abel equation
/ (x) =

l
a

x
(

0 < a < 1.

g (y) Ct dy,
Y)

X -

2. The Cauchy type integral equation
g(x)=/(x)+)..

l
a

b

g (y)

--dy,
y-x

where).. is a parameter.

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x

Preface

3. The extension
a (x) g (x) = J (x)

+)..

l -b

a

g (y)

y-x

dy ,

of the Cauchy equation. This is called the Carle man equation.
4. The integral equation of logarithmic kernel,

lb

In Ix

-

yl g(y) dy = J(x)

5. The Wiener-Hopf integral equation
g(x)+)..

1

00

K(x-y)g(y)dy=J(x)

The distinguishing feature of this equation is that the kernel is a
difference kernel and that the interval of integration is [0, (0).
We examine several variants and extensions of these equations, for example, on contours of the complex plane. Similarly, we present the solutions of double and triple integral equations. But unlike regular equations,
no general theory is available for singular integral equations, so that all
of the above-mentioned singular equations are studied on an ad-hoc basis.
We have therefore introduced generalized functions to provide a common
thread in our analysis of these equations.
The plan of the book is as follows.
In the first chapter we have included some reference material that will be
needed throughout the book. We present the basic principles of improper
integrals and singular integrals, as well as the fundamentals of Lebesgue
integration. We also consider two very important methods to assign a
value to a divergent integral, namely, the Cauchy principal value and the

Hadamard finite part. We also present the concepts of boundary values
of analytic functions and the various formulas related to this subject. We
introduce several spaces of functions and distributions to be used in our
studies. Finally, we give some ideas on transform analysis, considering in
particular, the Fourier and Laplace transforms.
The Abel integral equation is one of the simplest integral equations.
We present its solution in Chapter 2. We also study related equations.
The analysis is then extended to the case when the path of integration is
a contour in the complex plane. In addition to being very applicable, the

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Preface

Xl

Abel equation facilitates the solution of the Cauchy integral equation, as
we demonstrate in the next chapter.
Chapter 3 is devoted to the study of the Cauchy type integral equation.
We start by solving the Cauchy equation of the first kind on a real interval
and then extend the analysis to discuss the solution of the corresponding
equation of the second kind. Thereafter we solve the Cauchy type equation
in the complex plane, where we integrate along a contour, open or closed.
We consider the cases of contours that extend to infinity and the case of
disjoint intervals of integration. We finish the chapter by studying a very
important related integral equation, namely, the Hilbert equation.
The Carleman integral equation is an extension of the Cauchy type integral equation. We present its solution and many of its generalizations in
Chapter 4. We first study the equation over a finite real interval and then
extend the analysis to equations over contours in the complex plane. The

solution is achieved by introducing sectionally analytic functions, which
enables us then to transform the integral equation into interesting kinds of
boundary value problems, known as Riemann-Hilbert problems.
In our research we discovered that the distributional framework is very
effective for the study of singular integral equations and we present it in
Chapter 5. First we define various spaces of generalized functions and
then we present the distributional solutions of the integral equations of
Abel, Cauchy and Carleman type. In the process we introduce the concept
of the analytic representation of a distribution of compact support, which
is a very useful extension of the classical notion of analytic representation
introduced in Chapter 3. We also consider distributional dual and multiple
integral equations of the Cauchy type as well as distributional solutions of
these integral equations in the space of generalized periodic functions.
In Chapter 6 we present the distributional solution of the Carleman equation on the whole line, namely,
a (x)g(x)

+ -b (x)
7r

/00 -g -(y)d y = f (x)
-00 Y - x

,

where a (x) and b (x) are given smooth functions, while g (x) and f (x)
are distributions. The integral in the left side of this equation is the Hilbert
transform. We discover that this transform can be analyzed in certain suitable invariant subspaces of the space of tempered distributions S'. Thereafter we present asymptotic estimates for the Hilbert transform of test
functions and distributions. It turns out that in order to understand this
integral equation, we have to use the recently developed and rather inter-

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xii

Preface

esting mathematical apparatus of distributional asymptotic expansions and
the Cesaro behavior of generalized functions.
The singular integral equations with logarithmic kernels arise in several two-dimensional problems in mathematical physics, mechanics and
engineering. Accordingly, we devote Chapter 7 to the study of the main
singular integral equations of this type. We start with the simplest one and
then present the solution of more involved ones. Methods based on generalized functions developed in earlier chapters are put to very good use
in this chapter. Furthermore, we present the asymptotic behavior of the
eigenvalues of the basic integral equation with logarithmic kernel.
The solution of Wiener-Hopf type integral equations is presented in
Chapter 8. We begin by discussing the holomorphic Fourier transform
of both classical and generalized functions. Thereafter we present a mathematical technique for solving Wiener-Hopf integral equations of the second kind and give a rigorous justification of our analysis. Subsequently, we
show that equations of the first kind as well as the corresponding integrodifferential equations can be solved in the same fashion. Distributional solutions are also studied.
In the final chapter we study the dual and triple integral equations whose
kernels are trigonometric and Bessel functions. We present the classical
solutions of the integral equations of Beltrami and Titchmarsh type. The
operational calculus provided by the Erdelyi-K6ber operators helps us in
the process. Finally, we give the distributional solutions of the dual integral
equations of Titchmarsh type.
Numerous examples, illustrations, and exercises occur throughout the
book. Indeed, the book may be used in a one semester graduate course.
We hope that it will also benefit many researchers in the disciplines of
applied mathematics, mathematical physics and engineering.

Acknowledgments. Edwin Beschler, who retired from Birkhauser a few
years ago, has given us great encouragement in publishing our books in the
mathematical sciences. We take this opportunity to thank him for his many
years of helpful cooperation. We also thank Ann Kostant and Tom Grasso
of Birkhauser for continuing this cooperation, and to Elizabeth Loew for
the final preparation of this book.

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1
Reference Material

1.1

Introduction

In this chapter we give a brief summary of several topics and results that
are used throughout the book.
There are two possible ways to use this chapter. The reader may read
it all before the rest of the book, going as fast as suitable on well-known
topics. Alternatively, the reader could start with Chapter 2 and refer to this
first chapter whenever needed.
We start with a description of singular integral equations, weakly singular equations in particular, Fredholm integral equations, and Volterra integral equations, and of the concept of resolvent.
Then we consider the problem of giving meaning to various improper
integrals, both when the integrand is an unbounded function and when the
integrals are over infinite intervals. We give the classical ideas of absolute
and conditional convergence. We then proceed in Section 1.4 to study the
basic ideas of the Lebesgue integral and explain how this theory permits
us to consider several improper integrals. Next we study the concept of

Cauchy principal value for divergent integrals in Section 1.5 and the concept of Hadamard finite part of divergent integrals in Section 1.6. We also
give the definition and basic properties of the gamma function and the beta
function, and state several useful integral identities.
R. Estrada et al., Singular Integral Equations
© Birkhäuser Boston 2000

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2

1. Reference Material

In the last part of the chapter we mention the basic function spaces
used in the theory of integral equations. We also give the construction
of the basic spaces of distributions. We end by giving some ideas of integral transform analysis, considering in particular, the Fourier and Laplace
transforms.

1.2

Singular Integral Equations

In this book we shall discuss the integral equations
/(s)

g(s)

=

lb

= /(s) +

/(s)

=

is

g(s) = /(s)

+

(1.1)

K(s, t)g(t) dt,

lb

K(s, t)g(t) dt ,

(1.3)

K(s, t)g(t) dt,

is

(1.2)

K(s,t)g(t)dt.

(1.4)

The kernel K(s, t) and the function /(s) are given while g(s) is to be
evaluated. Equations (1.1) and (1.2) are Fredholm integral equations of
the first and second kind, respectively. Equations (1.3) and (1.4) are called
Volterra integral equations of the first and second kind, respectively.
If the domain of definition of the kernel is infinite, or if the kernel has a
singularity within its domain of definition, the integral equation is said to
be singular. Singular equations are our main focus in this book.
1~ certain cases, the kernel is only weakly singular as the singularity
miy be transformed away by a change of variable. For instance, a kernel
of the type
H(s, t)

= It-s la'

K(s, t)

0 < a < 1,

(1.5)

where H(s, t) is a bounded function, can be transformed to a kernel that is
bounded. Then, supposing that g(s) is bounded, we have

lb
a

K(s,t)g(t)dt=

is

H(s t)

' g(t)dt+
a (s - t)a

[b H(s ' t) g(t)dt.

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S

(t - s)a

(1.6)


1.3. Improper Integrals

= TJ Y , S - a = ~Y. Then

Consider the first integral and let s - t

i

s

= -y {~ H(s, s -

H(s, t) g(t) dt

TJY)g(s - TJY)TJY(1-a l - I dTJ.

10

a (s - t)a

3

The integral contains the factor TJy(l-al-l. The index y(l-a)-1 is positive
if a < 1 and y > (1 - a)-I, Accordingly, if a < 1, it is possible, by a
suitable transformation, to transform the singular integral into a regular
integral. An exactly similar treatment can be applied to the second integral
in (1.6). We consider a very important class of weakly singular integral
equations, the Abel-type integral equations, in Chapter 2.
Throughout most of the book, however, the kernels are more singular,
like the Cauchy kernel

1
K(s,t)=-,
s-t

(1.7)

whose integration requires the concept of principal value.
When we can find the solution of the integral equation
g(s)

= /(S)+A

in the form
g(s) = /(s)

then

f

f

+A

K(s,t)g(t)dt ,

(1.8)

f(s, t, A)/(t) dt,

(1.9)

r (s, t, A) is called the resolvent kernel of the integral equation.

1.3 Improper Integrals
Let / (x) be a bounded and integrable function over every interval [a, x],
where a is some fixed number and x is any number greater than a. The
infinite, or improper Riemann integral

1
00

is defined by the relation

1
a

00

/(y) dy

=

lim

1
x

x--+oo

a

/(y)dy

/(y) dy = lim F(x) - F(a) ,
x--+oo

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(1.10)

(1.11)


4

1. Reference Material

where F denotes any primitive of f. If the limit is finite, the integral is
said to converge, otherwise it is said to diverge. An integral of this type, in
which the range of integration becomes infinite but the integrand remains
bounded, is called an improper integral of the first kind. More generally,
we write

1:

f(y) dy =

x!~

1:

1

f(y) dy +

l

x!~oo x2 f(y) dy .

(1.12)

Improper integrals can be absolutely or conditionally convergent.
Absolute convergence. If Jaoo If(y)1 dy converges, then, since we have

II: f(Y)dyl
~ I: If(Y)1 dy, and limc.b-+oo I: If (Y)I dy =
oo

0, it follows

that Ja f(y) dy must also converge.

Conditional convergence. If Jaoo If (y) I dy diverges, then the integral
Loo f(y) dy may diverge, or it may converge. In the latter case, the convergence is said to be conditional.
Example 1. The integral

t
10

dx

sinx
x
is conditionally convergent but not absolutely convergent.
0< a < b, then
XJ

Indeed, if

lib Si:X dxl = 1_ co; xI: _ib c:~x dxl ~ ~,

and this last quantity tends to zero as a, b
verges.
On the other hand,

fOO Isin

10

x

xI dx = n-+oo
lim
Isin xI dx
10 x

L:
n-+oo

2: lim

Thus the integral con-

~ 00.

n-l

k=O 7r (

t1r

1
k

1

(k+1)1r

+ 1)

b

Isinxl

dx

2

L:--- n-+oo
(k + 1)
n-l

> lim

k=O 7r

=00 ,

so that the integral of the absolute value diverges.

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1.3. Improper Integrals

5

A useful test for the convergence or divergence of integrals is the comparison test .
Comparison test for improper integrals of the first kind. Let
be bounded integrable functions of x for a ~ x < 00 such that

I

and g

o ~ I(x) ~ g(x) ,
throughout this range. Then
(i) if Jaoo g(x) dx converges, so also does Loo I (x) dx.
(ii) if Jaoo I(x) dx diverges, so also does Jaoo g(x) dx.
A particularly useful application of this general test is obtained by taking
g(x)
For, if a > 0 and p

= x-p •

:f:. 1, then we get

1a

00

dx
-=lim
xP

b-+oo

lba

dx
x 1- p Ib
-=lim-xP
b-+oo 1 - P a

If P > 1, the integral converges to the value a 1- p / (1 - p). If p < 1, it
diverges to +00. While for p = 1,

1

00

a

dx
- = lim
X

lb

b-+oo a

dx
- = lim (In b - In a) =
X

b-+oo

00 ,

that is, it diverges to 00.
This analysis leads to a simple test, the p-test lor the improper integrals 01 the first kind. This test depends on the comparison theorem. Since
Jaoo 1/x P dx converges when p > 1, then for a non-negative function I(x)
for which I(x) ~ M /x P , M a fixed positive number, Jaoo I(x) dx also
converges.
The integral does not converge if there exists a positive number p ~ 1
and a fixed number M such that I (x) ::: M / x p • We just have to watch that
I (x) is non-negative in this interval; if it has positive and negative values,
then the areas might cancel out, and the integral may converge.
Let I be a bounded continuous function that is non-negative for all x :::
a. If there exists a number p such that limx-+oox P I(x) = A, then
(i)Jaoo I(x) dx converges if p > 1 and A is finite.
(ii)Loo I(x) dx diverges if p ~ 1 and A > 0 (possibly infinite).

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6

1. Reference Material

Now suppose that f (x) is a function that becomes unbounded as x approaches a in the interval [a, b]. We define

lb
a

f(x) dx == lim

lb

f(x) dx

e_O a+e

(1.13)

whenever this limit exists. Similarly, if f becomes unbounded as x approaches b in the interval [a, b], then we define

lb
a

f(x) dx == lim

lb-e

f(x) dx .

e_O a

(1.14)

Finally, ifxo is some point (a, b) and if f becomes unbounded in the neighf (x) dx is said to converge
borhood of Xo, then the improper integral
to the value

1:

l

lim

xo - el

+ lim

f(x) dx

el_O a

lb

e2- 0 XO+e2

f(x) dx ,

(1.15)

provided the limits exist independently. In other words, we require that
both the integrals
f (x) dx and
f (x) dx should converge.

Integrals such as these, in which the range of integration is finite but
the integrand becomes unbounded at one or more points of that range, are
called improper integrals of the second kind.
In case we have both an infinite range of integration and an integrand
that becomes unbounded within the range, we speak of an improper (Rie-

J:o

J!

mann) integral of the third kind.
In case the two integrals in (1.15) do not exist independently, we may
set el = e2 = e and then find that the unbounded parts of the two integrals
cancel each other out. This gives a certain finite answer, called the Cauchy
principal value of the divergent integral and written as
p.v.

lb
a

f(x) dx = lim

e_O

{l

a

xo e
-

f(x) dx

It is also written as

l

*b

a

f(x) dx

or

P

+ lb

lb

xo+e

f(x) dx

f(x)dx.

}.

(1.16)

(1.17)

We study principal value integrals in Section 1.5.
There is also a comparison test for improper integrals of the second kind.

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1.3. Improper Integrals

Comparison test for improper integrals of the second kind. Let
g be continuous for a ~ x ~ b and such that

o ~ I (x) ~ g(x) ,
Suppose also that both

J:

I

I

and

a~x~b.

and g become unbounded as x

~

I: I (x) dx, and
(ii) if I: I (x) dx diverges, then so does I: g(x) dx.
(i) if

7

a. Then

g(x) dx converges, then so does

Take
1

g(x)=-;

xP

if pi-I , then

t

10

dx =liml1dx =
1
Jl _ _
1 __ lim(el-P)
xP
£--.0 £ x P

(1 - p) x-1+p £ - 1 - P £--.0 1 - P .

It follows that this integral converges to 1/(1 - p) if p < 1, and that it
diverges to +00 if P > 1. For p = 1, we have

1
1

o

dx

-

X

.

= 11m

11 -dx = 11m
.
11
.
In x = In 1 - 11m In e = +00 .

£--.0 £

X

£

£--.0

£--.0

This leads us to the p-test. Let I be a continuous, non-negative function
for a ~ x ~ b and suppose that I becomes unbounded as x ~ a. If there
exists a number p such that
lim (x - a)P I(x)

x--.a

= A,

(1.18)

then
(i)

I: I (x) dx converges if p < 1 and A is finite,
(ii) I: I (x) dx diverges if p ~ 1 and A > 0 (possibly infinite).

There are two very important improper integrals that we now discuss,
the gamma and the beta functions.

1.3.1

The Gamma function

The gamma function is defined as
r(a)

=

1

00

x a - 1 e-x dx.

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(1.19)


8

1. Reference Material

For the integrand f(x) = xCl'-l e-x, x = 0 is a point of infinite discontinuity if a < 1. Thus we have to examine the convergence at x = 00 as
well as at x = O. Accordingly, we take any positive number greater than
zero, say 1, and examine the convergence of

1
00

and

f(X) dx,

at x = 0 and at x = 00 , respectively.
To establish the convergence at zero, we take g(x) = 1/x 1-CI' and find
that limx_o f(x)/g(x) = 1. Since fol g(x) dx = fol XCl'-1 dx converges if
and only if (1 - a) < 1, the original integral converges at x = 0 if only if
a> O.
To prove that it converges at 00, we take g(x) = 1/x2, and observe that
for any given a, we have for sufficiently large x, e > xCI'+! , or XCl'-l e-x <
1/x 2 • But
x- 2 dx converges, and hence

ft

also converges, and we have established the convergence of r(a) as defined by (1.19).
Integration by parts yields
r(a

+ 1) =

lim {b xCl'e-X dx
b-oo

10

= lim (_xCl'e-xl b+a

0 10(b XCl'-l e-x dX)

b_oo

=a (lim

b_oo

(b XCl'-l e-x dX)

10

= ar(a) ,

or
r(a

since bCl' / eb -+ 0 as b -+

00.

r(1)

+ 1) =

ar(a) ,

Using the fact that

=

1

00

e-X dx

= 1,

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(1.20)


1.3. Improper Integrals

9

relation (1.20) yields r(2) = lr(l) = 1, r(3) = 2r(2) = 2·1, r(4) =
3r(3) = 3 ·2· 1. Continuing in this way we have

r(n+l)=nl,

(1.21)

for any integer n 2: 0.
We began our discussion of the gamma function under the assumption
that a 2: 0, and we observed that the integral (1.19) does not exist if a = 0.
However, we can define rea) for negative values of a with the aid of this
integral if we set

rea)

=

r(a + 1)
a

°

.

(1.22)

Indeed, if -1 < a < 0, then
< a + 1 < 1, so that the right side
of equation (1.22) has a value, and the left side of (1.22) is defined to
have the value given by the right side. Similarly, if - 2 < a < -1, then
-1 < a + 1 < 0, so we can again use (1.22) to define r(a) on this interval
in terms of rea + 1), already defined in the previous step. This process can
be continued indefinitely.
It is convenient to allow a to take complex values. If ffie a > 0, then the
integral (1.19) converges. Ifffiea < 0, a =f:. 0, -1, -2, ... ,we may use
(1.22) to obtain the analytic continuation. The function r (a) becomes an
analytic function in C \ {O, -1, -2, ... }.
Observe that as a approaches 0, the function r (a) = r (a + 1) fa
behaves like 1/a. Thus a = is a simple pole with residue

°

Resa=o

r (a) =

1.

(1.23)

In particular,
lim rea) = 00,

a~O+

lim rea) = -00.

a~O-

From the way we have extended the gamma function for the negative values of a, it follows that it behaves in a similar way near all the negative
integers. Actually,

Resa=-k r (a)

(-1/

= J;! ,

for kEN.

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(1.24)


10

1. Reference Material

The gamma function enables us to give a meaning to the expression ex!
by setting

ex! = r(ex

+ 1),

for all values of ex except negative integers. This formula has its usual
(factorial) meaning when ex is a non-negative integer.
The gamma function r(ex) for half-integral values, i.e., ex = n + 1/2, is
very useful and is presented below. Indeed, for ex = 1/2, we have
(1.25)
By setting x

= 52,

or

When we change the double integral to polar coordinates
x

we have

= r cose,

y = r sine,

(r"2(1))2 10r/2 10(>0 e= 4

r

2

r dr de =

1T •

Thus
(1.26)
or (-1/2)! =

..fii.

Also

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1.3. Improper Integrals

11

Continuing this way we have
(n

+ 1/2)! =

(2n + 1)!
22n+1 , .Jii,

n.

(1.27)

and
(n -

(2n)!

1/2)! = -2-.Jii ,
2 nn!

(1.28)

for any non-negative integer n.

1.3.2

The Beta function

The beta function is defined by
{3(m, n) =

11

x m- 1(1 - x)n-1 dx ,

m > 0, n > O.

(1.29)

This integral is proper if m ?: 1 and n ?: 1. The point x = 0 is a point of
infinite discontinuity if m < 1, and the point x = 1 is a point of infinite
discontinuity if n < 1.
Let m < 1 and n < 1. We take any number, say 1/2, between 0 and 1
and examine the convergence of the improper integrals

t/2

10

x m- 1(1 - x)n-1 dx ,

0 and x = 1, respectively. For examining the convergence at
x = 0, we write I(x) = x m- 1(1 - x)n-1, and take g(x) = 1/x 1- m.
Then limx--.o I (x) / g (x) = 1. But the improper integral f01/2 g(x) dx =
at x

=

f01/21/xl-m dx converges at x = 0 if and only if 1 - m < 1, that is, if and

only if m > O. Thus, by the comparison test, the improper integral

is convergent at x = 0 if and only if m > O.
The convergence at x = 1 is established by writing I(x) = x m- 1(1 x)n-land taking g(x) = 1/(1 - x)l-n. Then limx--.l I (x) /g (x) = 1.

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12

1. Reference Material

Since J11/2 g(x) dx = J11/21/(1 - x)l-n dx is convergent if and only if
1 - n < 1, that is, if and only if n > 0, we deduce that

converges if and only if n > O.
The gamma and beta functions are connected by the relation
R

/J(m, n) =

f(m)r(n)
f(m

+ n)

(1.30)

.

One may use this formula to define the beta function for all complex numbers m, nEe \ to, -1, -2, ... }.

1.3.3 Another important improper integral
Another improper integral that we shall encounter is the integral

1

xa-1
--dx.
o x +1
00

(1.31)

It is easy to see that the given improper integral converges when 0 < ex <

1. To evaluate it, we split it as

1

xa-1
--dx=
o x+1
00

11

x a- 1
--dx+
x+1

0

1

00

1

x a- 1
--dx.
x+1

(1.32)

By means of the substitution y = 1jx, we observe that

f

oo

1

xa-1
--dx=
x+1

11

y-a
--dy=
y+l

0

11
0

x-a
--dx.
x+1

Thus, relation (1.32) becomes

1
1

11 (X

xa-1
--dx =
o x+l
00

=

1

o

0

(x a- 1 +x-a)

(

a- 1 + x-a)
dx
x+l

L (-xl + (-It+1-+1
n

xn+1 )

k=O

n
=L<-1l

k=O

X

(1
1)
--+
+Rn.
+
+
k

ex

k

1 - ex

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dx


1.3. Improper Integrals

13

where

and this last expression tends to 0 as n --+ 00. Hence

(>0

10

a
x - 1 dx

x+l

= I:(-I)k
k=O

By substituting x = y / (y

(_1_ + 1 )=
k+a

10 1x a - 1(1 -

we find that
1

o

_rr_.
sinarr

(1.33)

+ 1) in the relation

{J(a,1 - a) =

1

k+l-a

x a- 1(1 - x)-a dx =

100
0

X)-a dx,

ya-1
rr
- - dy = -.-- .
y+l
smarr

(1.34)

Combining (1.30) and (1.34) yields
rr
.
smarr

{3(a, 1 - a) = r(a)r(l - a) = -.-

(1.35)

Similarly, it can be proved that
p.v.

1

00

xa-1dx

o

I-x

= rr cotarr .

(1.36)

1.3.4 A few integral identities
Consider the integral
1=

t (U - S)a

10

-s-

ds

(u-s)(s-t) ,

and set u - s = vs in it, so that

1--

1

00

va-1 dv

o vt - (u - t)

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(1.37)