Asymptotic Approximations of Integrals

✍ Scribed by Wong, Roderick

• Publisher: Academic Press
• Year: 1989
• Tongue: English
• Leaves: 552
• Series: Computer Science and Scientific Computing
• Edition: First
• Category: Library

✦ Synopsis

Asymptotic methods are frequently used in many branches of both pure and applied mathematics, and this classic text remains the most up-to-date book dealing with one important aspect of this area, namely, asymptotic approximations of integrals. In this book, all results are proved rigorously, and many of the approximation formulas are accompanied by error bounds. A thorough discussion on multidimensional integrals is given, with references provided. Asymptotic Approximations of Integrals contains the 'distributional method', not available elsewhere. Most of the examples in this text come from concrete applications. Since its publication twelve years ago, significant developments have occurred in the general theory of asymptotic expansions, including smoothing of the Stokes phenomenon, uniform exponentially improved asymptotic expansions, and hyperasymptotics. These new concepts belong to the area now known as 'exponential asymptotics'. Expositions of these new theories are available in papers published in various journals, but not yet in book form

✦ Table of Contents

Content: Front Cover
Asymptotic Approximations of Integrals
Copyright Page
Dedication
Table of Contents
Preface
Chapter I. Fundamental Concepts ofAsymptotics
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
Chapter 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 NotesChapter 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
Chapter 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. 4. Hilbert Transforms5. Laplace and Fourier Transforms Near the Origin
6. Fractional Integrals
7. The Method of Regularization
Exercises
Supplementary Notes
Chapter VII. Uniform AsymptoticExpansions
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. LegendreFunction
Exercises
Supplementary Notes
Chapter VIII. Double Integrals
1. Introduction
2. Classification of Critical Points
3. Local Extrema
4. Saddle Points. 5. A Degenerate Case6. 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
Chapter IX. Higher DimensionalIntegrals
1. Introduction
2. Stationary Points
3. Points of Tangential Contact
4. Degenerate Stationary Point
5. Laplace's Approximation inRn
6. Multiple Fourier Transforms
Exercises
Supplementary Notes
Bibliography
Symbol Index
Author Index
Subject Index.

✦ Subjects

Integrals;Approximation theory;Asymptotic expansions;Intégrales;Théorie de l'approximation;Développements asymptotiques;MATHEMATICS -- Calculus;MATHEMATICS -- Mathematical Analysis;Approximation, théorie de l';Approximation;Asymptotische Methode;Integral