Asymptotic Expansions of Integrals

Excellent introductory text, written by two experts, presents a coherent and systematic view of principles and methods. Topics include integration by parts, Watson's lemma, LaPlace's method, stationary phase, and steepest descents. Additional subjects include the Mellin transform method and less elementary aspects of the method of steepest descents. 1975 edition.

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