Asymptotic Approximations of Integrals

PrefaceI 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 NotesII 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 NotesIII 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 TransformsIV 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 NotesV 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 NotesVI 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 NotesVII 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 NotesVIII 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 NotesIX 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 NotesBibliographySymbol IndexAuthor IndexSubject Index