Synopses & Reviews
Asymptotic analysis, that branch of mathematics devoted to the study of the behavior of functions within chosen limits, was once thought of more as a specialized art than a necessary discipline. Now, a solid foundation in the theory and technique of asymptotic expansion of integrals is at the heart of the education of every student concentrating in applied mathematics. It is also an invaluable asset to scientists in many other fields.This excellent introductory text, written by two experts in the field, offers students of applied mathematics — and researchers and workers in other fields — a coherent and systematic presentation of the principles and methods of asymptotic expansions of integrals. Students will find each of the nine chapters useful and easy to comprehend; each begins with elementary material or informal introductions and concludes with an abundant selection of applicable problems and exercises. All that is needed is a basic understanding of calculus, differential equations, and complex variables.Practiced users of asymptotics will find the work a valuable reference with an extensive index locating all the functions covered in the text and every formula associated with the major techniques. Subjects include integration by parts, Watson's lemma, Laplace's method, stationary phase, and steepest descents. Also treated are the Mellin transform method and less elementary aspects of steepest descent. Each chapter is carefully illustrated with helpful diagrams and tables.Used for years as a text in classrooms throughout the country, the book has been revised and corrected for this inexpensive paperback edition. Any student or teacher looking for a suitable text for a year's or semester's course in asymptotics will value this affordable volume as the only comprehensive introduction available; scientists and researchers will undoubtedly refer to it again and again.
Unabridged, corrected Dover (1986) republication of the edition published by Holt, Rinehart and Winston, New York, 1975.
Synopsis
Coherent, systematic coverage of standard methods: integration by parts, Watson's lemma, LaPlace's method, stationary phase and steepest descents. Also treated are the Mellin transform method and less elementary aspects of the method of steepest descents. Abundant exercises. 1975 edition.
Synopsis
Excellent introductory text by two experts presents a coherent, systematic view of principles and methods. Topics include integration by parts, Watson's lemma, Laplace's method, stationary phase, and steepest descents. 1975 edition.
Synopsis
Excellent introductory text by two experts presents a coherent, systematic view of principles and methods. Topics include integration by parts, Watson's lemma, LaPlace's method, stationary phase, and steepest descents. 1975 edition.
Table of Contents
CHAPTER 1. Fundamental Concepts
1.1. Introduction
1.2 Order Relations
1.3. Asympototic Power Series Expansions
1.4. Asymptotic Sequences and Asymptotic Expansions of Poincaré Type
1.5. Auxiliary Asymptotic Sequences
1.6. Complex Variables and Stokes Phenomenon
1.7 Operations with Asymptotic Expansions of Poincaré Type
1.8. Exercises
References
CHAPTER 2. Asymptotic Expansions of Integrals: Preliminary Discussion
2.1. Introduction
2.2. The Gamma and Incomplete Gamma Functions
2.3. Integrals Arising in Probability Theory
2.4. Laplace Transform
2.5. Generalized Laplace Transform
2.6. Wave Propagation in Dispersive Media
2.7. The Kirchhoff Method in Acoustical Scattering
2.8. Fourier Series
2.9. Exercises
References
CHAPTER 3. Integration by Parts
3.1. General Results
3.2. A Class of Integral Transforms
3.3. Identification and Isolation of Critical Points
3.4. An Extension of the Integration by Paris Procedure
3.5. Exercises
References
CHAPTER 4. h-transforms with Kernels of Monotonic Argument
4.1. Laplace Transforms and Watson's Lemma
4.2. Results on Mellin Transforms
4.3. Analytic Continuation of Mellin Transforms
4.4. Asymptotic Expansions for Real ?
4.5. Asymptotic Expansions for Real ?: Continuation
4.6. Asymptotic Expansions for Small Real ?
4.7. Asymptotic Expansions for Complex ?
4.8. Electrostatics
4.9. Heat Conduction in a Nonlinearly Radiating Solid
4.10. Fractional Integrals and Integral Equations of Abel Type
4.11. Renewal Processes
4.12. Exercises
References
CHAPTER 5. h-Transforms with Kernals of Nonmonotonic Argument
5.1. Laplace's Method
5.2. Kernels of Exponential Type
5.3. Kernels of Exponential Type: Continuation
5.4. Kernels of Algebraic Type
5.5. Expansions for Small ?
5.6. Exercises
References
CHAPTER 6. h-Transforms with Oscillatory Kernels
6.1. Fourier Integrals and the Method of Stationary Phase
6.2. Further Results on Mellin Transforms
6.3. Kernels of Oscillatory Type
6.4. Oscillatory Kernels: Continuation
6.5. Exercises
References
CHAPTER 7. The Method of Steepest Descents
7.1. Preliminary Results
7.2. The Method of Steepest Descents
7.3. The Airy Function for Complex Agrument
7.4. The Gamma Function for Complex Argument
7.5. The Klein-Gordon Equation
7.6. The Central Limit Theorem for Identically Distributed Random Variables
7.7. Exercises
References
CHAPTER 8. Asymptotic Expansions of Multiple Integrals
8.1. Introduction
8.2. Asymptotic Expansions of Double Integrals of Laplace Type
8.3. Higher-Dimensional Integrals of Laplace Type
8.4. Multiple Integrals of Fourier Type
8.5. Parametric Expansions
8.6. Exercises
References
CHAPTER 9. Uniform Asymptotic Expansions
9.1. Introduction
9.2. Asymptotic Expansion of Integrals with Two Nearby Saddle Points
9.3. Underlying Principles
9.4. Saddle Point near on Amplitude Critical Point
9.5. A Class of Integrals That Arise in the Analysis of Precursors
9.6. Double Integrals of Fourier Type
9.7. Exercises
References
Appendix
General References
Index