Asymptotic Approximations of Integrals - 1st Edition - ISBN: 9780127625355, 9781483220710

Asymptotic Approximations of Integrals

1st Edition

Computer Science and Scientific Computing

Authors: R. Wong
Editors: Werner Rheinboldt Daniel Siewiorek
eBook ISBN: 9781483220710
Imprint: Academic Press
Published Date: 28th September 1989
Page Count: 556
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Description

Asymptotic Approximations of Integrals deals with the methods used in the asymptotic approximation of integrals. Topics covered range from logarithmic singularities and the summability method to the distributional approach and the Mellin transform technique for multiple integrals. Uniform asymptotic expansions via a rational transformation are also discussed, along with double integrals with a curve of stationary points. For completeness, classical methods are examined as well.

Comprised of nine chapters, this volume begins with an introduction to the fundamental concepts of asymptotics, followed by a discussion on classical techniques used in the asymptotic evaluation of integrals, including Laplace's method, Mellin transform techniques, and the summability method. Subsequent chapters focus on the elementary theory of distributions; the distributional approach; uniform asymptotic expansions; and integrals which depend on auxiliary parameters in addition to the asymptotic variable. The book concludes by considering double integrals and higher-dimensional integrals.

This monograph is intended for graduate students and research workers in mathematics, physics, and engineering.

Table of Contents


Preface

I Fundamental Concepts of Asymptotics

1. What Is Asymptotics?

2. Asymptotic Expansions

3. Generalized Asymptotic Expansions

4. Integration by Parts

5. Watson's Lemma

6. The Euler-Maclaurin Summation Formula

Exercises

Supplementary Notes

II Classical Procedures

1. Laplace's Method

2. Logarithmic Singularities

3. The Principle of Stationary Phase

4. Method of Steepest Descents

5. Perron's Method

6. Darboux's Method

7. A Formula of Hayman

Exercises

Supplementary Notes

III Mellin Transform Techniques

1. Introduction

2. Properties of Mellin Transforms

3. Examples

4. Work of Handelsman and Lew

5. Remarks and Examples

6. Explicit Error Terms

7. A Double Integral

Exercises

Supplementary Notes

Short Table of Mellin Transforms

IV The Summability Method

1. Introduction

2. A Fourier Integral

3. Hankel Transform

4. Hankel Transform (Continued)

5. Oscillatory Kernels: General Case

6. Some Quadrature Formulas

7. Mellin-Barnes Type Integrals

Exercises

Supplementary Notes

V Elementary Theory of Distributions

1. Introduction

2. Test Functions and Distributions

3. Support of Distributions

4. Operations on Distributions

5. Differentiation of Distributions

6. Convolutions

7. Regularization of Divergent Integrals

8. Tempered Distributions

9. Distributions of Several Variables

10. The Distribution rλ

11. Taylor and Laurent Series for rλ

12. Fourier Transforms

13. Surface Distributions

Exercises

Supplementary Notes

VI The Distributional Approach

1. Introduction

2. The Stieltjes Transform

3. Stieltjes Transform: An Oscillatory Case

4. Hubert Transforms

5. Laplace and Fourier Transforms near the Origin

6. Fractional Integrals

7. The Method of Regularization

Exercises

Supplementary Notes

VII Uniform Asymptotic Expansions

1. Introduction

2. Saddle Point Near a Pole

3. Saddle Point near an Endpoint

4. Two Coalescing Saddle Points

5. Laguerre Polynomials I

6. Many Coalescing Saddle Points

7. Laguerre Polynomials II

8. Legendre Function Pn-m(Cosh z)

Exercises

Supplementary Notes

VIII Double Integrals

1. Introduction

2. Classification of Critical Points

3. Local Extrema

4. Saddle Points

5. A Degenerate Case

6. Boundary Stationary Points

7. Critical Points of the Second Kind

8. Critical Points of the Third Kind

9. A Curve of Stationary Points

10. Laplace's Approximation

11. Boundary Extrema

Exercises

Supplementary Notes

IX Higher Dimensional Integrals

1. Introduction

2. Stationary Points

3. Points of Tangential Contact

4. Degenerate Stationary Point

5. Laplace's Approximation in Rn

6. Multiple Fourier Transforms

Exercises

Supplementary Notes

Bibliography

Symbol Index

Author Index

Subject Index

Details

No. of pages:
556
Language:
English
Copyright:
© Academic Press 1989
Published:
Imprint:
Academic Press
eBook ISBN:
9781483220710

About the Author

R. Wong

About the Editor

Werner Rheinboldt

Daniel Siewiorek

Affiliations and Expertise

Carnegie-Mellon University