Asymptotic Approximations of Integrals. Computer Science and Scientific Computing

549 Pages·1989·16.47 MB·English

by  R. Wong

Asymptotic Approximations
of Integrals
R. Wong
Department of Applied Mathematics
The University of Manitoba
Winnipeg, Manitoba
Canada
ACADEMIC PRESS, INC.
Harcourt Brace Jovanovich, Publishers
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Copyright © 1989 by Academic Press, Inc.
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ACADEMIC PRESS, INC.
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United Kingdom Edition published by
ACADEMIC PRESS INC. (LONDON) LTD.
24-28 Oval Road, London NW1 7DX
Library of Congress Cataloging-in-Publication Data
Wong, R. (Roderick), Date —
Asymptotic approximations of integrals / R. Wong.
p. cm. (Computer science and scientific computing)
Includes bibliographies and index.
ISBN 0-12-762535-6
1. Integrals. 2. Approximation theory. 3. Asymptotic expansions.
I. Title. II. Series.
QA311.W65 1989
515. 4'3 dcl9
89-137
CIP
Printed in the United States of America
89 90 91 92 9 8 7 6 5 4 3 2 1
To
my mother
who gave me
life and education
Preface
Asymptotic approximation is an important topic in applied analysis,
and its applications permeate many fields in science and engineering
such as fluid mechanics, electromagnetism, diffraction theory, and
statistics. Although it is an old subject, dating back to the time of
Laplace, new methods and new applications continue to appear in
various publications. There are now several excellent books on this
subject, and, in particular, the one by F. W. J. Olver deserves special
mention. However, most of these books were written more than 15
years ago, and Olver's book stresses more the differential equation side
of asymptotic theory. There is now a need to provide an up-to-date
account of methods used in the other main area of asymptotic theory,
namely, asymptotic approximation of integrals. The purpose of this
book is precisely to fulfil this need. Many of the results appear for the
first time in book form. These include logarithmic singularities, Mellin
transform technique for multiple integrals, summability method, distri-
butional approach, uniform asymptotic expansions via a rational
transformation, and double integrals with a curve of stationary points.
For completeness, classical methods are also discussed in detail.In
this sense, the book is self-contained. Furthermore, all results are
XI
XU Preface
proved rigorously and accompanied by error bounds whenever
possible.
The book presupposes that the reader has a thorough knowledge of
advanced calculus and is familiar with the basic theory of complex
variables. It can be used either as a text for graduate students in
mathematics, physics, and engineering, or as a reference tool for
research workers in these, and other, areas. As a text, it is suitable for a
two-semester course meeting three hours per week, but it can also be
used for a one-semester course. For instance, Chapters I to IV and parts
of Chapter VII would constitute such a course. Each chapter is self-
contained in order to render it accessible to the casual peruser.
Each chapter has an extensive set of exercises, many of which are
accompanied by hints for their solution. However, the development of
the material in the text does not depend on the exercises, and omission
of some or all of them does not destroy the continuity of the presenta-
tion. Nevertheless, students are strongly advised to read through the
exercises, since some of them provide important extensions of the
general theory, while others supply completely new results.
Chapters I and II cover classical techniques in the asymptotic
evaluation of integrals. More recent methods are introduced in
Chapters III, IV, and VI. In Chapter V, a short introduction to
distribution theory is presented, and almost all of the results given in
this chapter are used later. Integrals which depend on auxiliary
parameters in addition to the asymptotic variable are discussed in
Chapter VII. Finally, Chapters VIII and IX are devoted to multidimen-
sional integrals.
A short section titled "Supplementary Notes'' appears at the end of
each chapter, where additional references can be found. Some of these
supply sources of material presented, and some pertain to more recent
books or papers on closely related topics. Since mention to the
references Erdélyi et al (1953, 1953b), Olver (1974a), and Watson (1944)
for properties of special functions is made frequently throughout the
book, we have omitted their occurrences from the author index.
The writing of the manuscript began when I was a Killam Research
Fellow (1982-1984). The preparation of the book has been facilitated by
both the Killam Foundation and the Natural Sciences and Engineering
Council of Canada. To these agencies I am most grateful.
I must express my special thanks to Professor F. W. J. Olver, who has
read the entire manuscript as well as the page proofs, offered numerous
Preface χιιι
suggestions, and corrected innumerable errors, mathematical as well
as linguistic. Without his generous advice and constant encourage-
ment, this book would have never been written. I am indebted to my
colleague Professor J. P. McClure, and to Professor Qu Chong-kai of
Tsinghau University, both of whom read and commented on the
manuscript in its preliminary stages. Thanks are also due to Carol
Plumridge for her excellent job of typing, and to my student, Tom Lang,
for proofreading various parts of the typescript. Finally, my deep
gratitude goes to my wife, Edwina, for reading the page proofs and for
her patience and understanding during the preparation of the manu-
script.
I
Fundamental Concepts of
Asymptotics
1. What Is Asymptotics?
In analysis and applied mathematics, one frequently comes across
problems concerning the determination of the behavior of a function as
one of its parameters tends to a specific value, or of a sequence as its
index tends to infinity. The branch of mathematics that is devoted to
the investigation of these types of problems is called asymptotics. Thus,
for instance, results such as
log n\ ~ (n + |) log n — n + \ log 2π, (1.1)
H = 1 + - + - + ·· + - ~ log n, (1.2)
n
Δ o Tí
and
1 f»|BÍn(n + j)¿|^ 4
L = - ^ -— dt - -g log n, (1.3)
n
π J sin |£ 7T
0
are all part of this subject. Equation (1.1) is known as the Stirling
formula; the numbers H are called the harmonic numbers and often
n
1
2 I Fundamental Concepts of Asymptotics
occur in the analysis of algorithms (Greene and Knuth 1981); and the
numbers L in (1.3) are called the Lebesgue constants in the theory of
n
Fourier series. The twiddle sign ~ is used to mean that the quotient of
the left-hand side by the corresponding right-hand side approaches 1 as
n->oo. Formulas such as those in (1.1)-(1.3) are called asymptotic
formulas or asymptotic equalities.
Results in (1.1)-(1.3) are all easy to derive and can be found in books
on elementary analysis; see, for example, Rudin (1976). However, on
many occasions, information given by an asymptotic formula is insuffi-
cient and higher term approximations are required. The situation here
is very much akin to the one in which the approximation obtained from
the mean-value theorem is not sufficient and the use of Taylor's
formula with remainder becomes necessary. Higher-term approxima-
tions for log n\, H , and L are given by
n n
( 1 \ 1 °° fí
„ ^i „-„ - og2, z ¿;v- <">
+ og + 1 + o(2s+1)(
n 1 1 °° 7?
<L5)
*·-.?, ¡r'-'-^s-.C.iss*
and
L ^ - . j l o g^ + D + ^ - ^ ^ - ^l (1.6)
where B denotes the Bernoulli numbers defined by
'22ss
Z ^O O Z"1- 7
(17)
Ί-.?ΣΛ "Π ·
n =0
and
log m
A) = 2 Σ 2 , +21og2 + y = 2.441..., (1.8)
; A4m2 - 1
1
(l-22s"1)ß r, ^, (-1)"
2s 1- Σ y-17t^B n 2m s > 1, (1.9)
: (2m)! "22mm'
x
y being the Euler constant. Note that the series (1.4)-(1.6) are all
divergent; we have extended the symbol ~ to mean that any partial
sum of any one of these is an approximation of the corresponding left-
1. What Is Asymptotics? 3
hand side with an error that is of the same order of magnitude as the
first neglected term. The results are, in fact, more precise than this;
that is, one can show that the remainder due to truncation of any one
of these series has the property that it is numerically less than, and has
the same sign as, the first term omitted. These results are not easy to
obtain, and they form typical examples in books on asymptotics. A
proof of (1.4) is outlined in Ex. 19, and a proof of (1.5) is given in Section
6. The result in (1.6) is due to Watson (1930), and his proof is reproduced
in Section 6.
The present subject of asymptotics can be divided into three main
areas. The first area deals with functions that are expressible in the
forms of definite integrals or contour integrals. A typical example in
this area is given by the integral
/„= (" <p(x)[f(x)]n dx,
Ja
where φ(χ) and f(x) are continuous functions defined on the interval
[a, b] and f(x) is positive there. Long ago, Laplace made the observation
that the major contribution to the integral I should come from the
n
neighborhoods of the points where f(x) attains its greatest value.
Furthermore, he showed that if f(x) attains its maximum value only at
the point ξ in (a, b) where /*'(£) = 0 and /"(<!;) < 0, then
( —2πΊ1/2
This formula is now known as the Laplace approximation. For an
excellent introduction to the topics in this area, we refer to the book by
Copson (1965).
The second area in asymptotics is concerned with solutions to
differential equations. The best known equation here is probably
y" + [λ2α(χ) + b(x)]y = 0, (1.11)
where λ is a large positive parameter and a(x) > 0 in [x , xj. Liouville
0
and Green, simultaneously and independently, showed that equation
(1.11) has two linearly independent solutions, which behave asymptoti-
cally like
y± (x) ~ , exp ± ιλ a1/2(x) dx , as λ -► oo. (1.12)
aΛ1,Ι4Α(/x) !
4 I Fundamental Concepts of Asymptotics
Formula (1.12) has been known as the WKB approximation; it is only
recently that (1.12) is called, and rightly so, the Liouville-Green
approximation. For a definitive work on this area of asymptotics, see
Olver (1974a).
The third area in asymptotics is connected with enumeration prob-
lems. A typical example is the following: Let d denote the number of
n
partitions of an n-element set (e.g., d = 1, d = 2, d = 5, d = 15). It is
x 2 3 4
known that the exponential generating function of these numbers is
exp(ez — 1), i.e.,
00 d
exp(e2-1)= Σ ~^ζη· (1-13)
The problem here is to obtain the asymptotic behavior of d as n -► oo.
n
From a formula of Hayman (see Chapter II, Section 7), we have
d ^ « Ρ ^η + ^ " 1 - 1 ) - 1 ^ (1.14)
V Tn + 1
where r is the root of the equation
n
r exp(r) = n. (1.15)
For a survey of the methods in this area of asymptotics, see Bender
(1974).
The asymptotics of the first area will occupy a central portion of the
present book. Since problems in the third area are often related to
problems in the first area, some of the important methods in this area
will also be mentioned.
2. Asymptotic Expansions
In 1886, Poincaré introduced the notion of an asymptotic expansion.
This concept enables one to manipulate a large class of divergent series
in much the same way as convergent power series. Moreover, it enables
one to obtain numerical as well as qualitative results for many
problems. The divergent series in (1.4)-(1.6) are all special examples of
asymptotic expansions. Before giving a precise definition, let us first
recall the O- and o-symbols introduced by Landau.