Table Of ContentApplied
Mathematical
Sciences
EDITORS
Fritz John Joseph P. LaSalle Lawrence Sirovich
Courant Institute of Division of Division of
Mathematical Sciences Applied Mathematics Applied Mathematics
New York University Brown University Brown University
New York, N.Y. 10003 Providence, R.I. 02912 Providence, R.I. 02912
EDITORIAL STATEMENT
The mathematization of all sciences, the fading of traditional scientific boun
daries, the impact of computer technology, the growing importance of mathematical
computer modelling and the necessity of scientific planning all create the need both
in education and research for books that are introductory to and abreast of these
developments.
The purpose of this series is to provide such books, suitable for the user of
mathematics, the mathematician interested in applications, and the student scientist.
In particular, this series will provide an outlet for material less formally presented
and more anticipatory of needs than finished texts or monographs, yet of immediate
interest because of the novelty of its treatment of an application or of mathematics
being applied or lying close to applications.
The aim of the series is, through rapid publication in an attractive but in
expensive format, to make material of current interest widely accessible. This
implies the absence of excessive generality and abstraction, and unrealistic ideali
zation, but with quality of exposition as a goal.
Many of the books will originate out of and will stimulate the development
of new undergraduate and graduate courses in the applications of mathematics. Some
of the books will present introductions to new areas of research, new applications
and act as signposts for new directions in the mathematical sciences. This series may
serve as an intermediate stage of the publication of material which, through
exposure here, will be further developed and refined and appear later in the Mathe
matics in Science Series of books in applied mathematics also published by Springer
Verlag and in the same spirit as this series.
MANUSCRIPTS
The Editors welcome all inquiries regarding the submission of manuscripts
for the series. Final preparation of all manuscripts will take place in the editorial
offices of the series in the Division of Applied Mathematics, Brown University, Prov
idence, Rhode Island.
Published by SPRINGER SCIENCE+BUSINESS MEDIA, LLC
L. Sirovich
Techniques of
Asymptotic
Analysis
With 23 Illustrations
Springer Science+Business Media, LLC 1971
Lawrence Sirovich
Division of Applied Mathematics
Brown University
Providence, Rhode Island
AU rights reserved
No part of this book may be translated or reproduced in any form
without written permission from Springer-Verlag_
© 1971 by Springer Science+Business Media New York
Originally published by Springer-Verlag New York • Heidelberg • Berlin in 1971
Softcover reprint ofthe hardcover Ist edition 1971
Library of Congress Catalog Card Number 70-149141
ISBN 978-0-387-90022-3 ISBN 978-1-4612-6402-6 (eBook)
DOI 10.1007/978-1-4612-6402-6
I
Applied Mathematical Sciences Volume 2
PREFACE
These notes originate from a one semester course which forms part of the
"Math Methods" cycle at Brown. In the hope that these notes might prove useful for
reference purposes several additional sections have been included and also a table
of contents and index.
Although asymptotic analysis is now enjoying a period of great vitality,
these notes do not reflect a research oriented course. The course is aimed toward
people in applied mathematics, physics, engineering, etc., who have a need for
asymptotic analysis in their work. The choice of subjects has been largely dictated
by the likelihood of application. Also abstraction and generality have not been
pursued. Technique and computation are given equal prominence with theory.
Both rigorous and formal theory is presented -- very often in tandem. In
practice, the means for a rigorous analysis are not always available. For this
reason a goal has been the cultivation of mature formal reasoning. Therefore,
during the course of lectures formal presentations gradually eclipse rigorous
presentations. When this occurs, rigorous proofs are given as exercises or in the
case of lengthy proofs, reference is made to the Reading List at the end.
The Reading List contains a number of books for further reading. Among
these are included those books which have influenced me in the preparation of this
course. Most noteworthy, in this respect, are the treatments by Friedrichs, Erdelyi,
and Dieudonne. In the case of Professor Friedrichs my debt goes back to my student
days when I took a course with him on asymptotic analysis. This has had a lasting
effect on me in my scientific work.
Finally, I wish to acknowledge the invaluable assistance of T. H. Chong
and C. Huo in the preparation of these notes. I am also indebted to Ezora Fonseca,
Katherine MacDougall and Sandra Spinacci for their careful typing of my handwritten
notes and to Eleanor Addison for her excellent preparation of my sketchs.
Providence, Rhode Island Lawrence Sirovich
December, 1970
v
Notation:
The material is divided into three chapters and each chapter into sections.
Thus, on referring, for example, to Section 3.5 we mean Chapter 3, Section 5.
Equations are individually numbered in each section, for example, (3.5.1) refers to
the first equation of Section 3.5. Theorems are numbered in the same way but with
out punctuation marks. A bracketed number such as [1) refers to a book on the
Reading List, p. 300.
TABLE OF CONTENTS
PREFACE v
CHAPTER 1. ASYMPTOTIC SEQ,UENCES .AND ASYMPTOTIC DEVELOPMENT
OF A FUNCTION
1.1. Notation and Definition l
1.2. Operations with Asymptotic Expansions 11
Asymptotic Integration 17
Differentiation 21
1.3. Same Remarks on the Use of Asymptotic Expansions 24
1.4. Summation of Asymptotic Expansions ••••••• 28
CHAPTER 2. THE ASYMPTOTIC DEVELOPMENT OF A FUNCTION DEFINED
BY AN INTEGRAL
2.1. Elementary Analytic Methods 38
Analytic Continuation of Functions Defined
by an Integral 38
Integration by Parts . • 4o
Asymptotic Evaluation of Indefinite Integrals 44
Asymptotic Evaluation of Integrals of the
Form f X f(x,t)dt •••• 54
2.2. Laplace and Fourier Transforms at Infinity 62
Watson' s Lemma 65
Fourier Integrals 74
2.3. Laplace's Formula and Its Generalization 80
2.4. Kelvin's Formula and Generalizations 86
2.5. Integrals of the Type f~((~))G(x,t)dt 95
,.,
a~
Generalized Laplace Formula 96
Generalized Kelvin's Formula 100
Dispersive Wave Propagation 102
2.6. Method of Steepest Descents and the
Saddle Point Formula . • . • . • 105
Saddle Point Method for a Complex
Large Parameter • • . • • • • . • 115'
The Complete Asymptotic Development 117
Application to Bessel Functions 122
2.7. Applications of the Saddle Point Method 126
The Airy Integral 126
vii
A Generalization of the Airy Integral 134
2.8. Multidimensional Integrals: Part I.
Laplace, Kelvin, and Related Formulas 136
Multidimensional Integrals: Part II.
. . . . .
Many Parameters 148
~
Complete Asymptotic Development 155
2.10. Asymptotic Eyaluation of Integrals
lpvolving Non-Uniformities ; • • • 164
Integrals Containing a Global Maximum
Near an Endpoint • 165
Neighboring Saddle Points 170
2.11. Miscellaneous 176
Laplace Transforms in the Neighborhood of
the Origin ••••••••..• 176
Fourier Transforms at the Origin 181
Bramwich Integrals at Infinity • 182
Br~wich Integrals at t~e Origin 186
CHAPTER 3 . LINEAR OBI>INARY DIFFERENTIAL EQUATIONS
3 .0. Introduction • • • • • . • . . 189
3.1. Same Topics in Matrix Analysis 192
Applications to Ordinary Differential Equations 195
3.2, Matrix Theory - Continued 201
Functions of Matrices 207
Dunford-Taylor Integral 210
Construction of a Function of a Matrix 211
3.3 Linear Ordinary Differential Equations with
1
Constant Coefficients . • • • • . • • • . • 219
3.4. Classification and General Properties of Ordinary
Differential Equations in the Neighborhood of
Singular Points 226
Circuit Relations 230
Singular Points of an nth Order Scaler Ordinary
Differential Equations . • • • . 232
Solutions in the Neighborhood of Infinity 234
dX
The Equation z...:::::. =A X ••••••• 235
dz ~av
3~5. Linear Ordinary Differential Equations with
Regular Singular Points • • . • • 240
The Case of a Scaler Ordinary Differential
Eq~ation with a Regular Singular Point 254
Method of Frbbenius 256
3.6. Irregular ~ingul~r Points 259
viii
Scaler Ordinary Differential EQuations 278
Second Order EQuations . • • 284
3.7. Ordinary Differential EQuations Containing
a Large Parameter
Formal Solution
Turning or Transition Points
Connection Formulas
Langer's Uniform Method
READING LIST 300
INDEX 301
ix
