Wong / Rheinboldt / Siewiorek Asymptotic Approximations of Integrals

Computer Science and Scientific Computing
1. Auflage 2014
ISBN: 978-1-4832-2071-0
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 556 Seiten, Web PDF

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.

Autoren/Hrsg.

Wong, R.

Rheinboldt, Werner

Siewiorek, Daniel

Inhaltsverzeichnis

1;Front Cover;1
2;Asymptotic Approximations of Integrals;4
3;Copyright Page;5
4;Table of Contents;8
5;Dedication;6
6;Preface;12
7;Chapter I. Fundamental Concepts of
Asymptotics;16
7.1;1. What Is Asymptotics?;16
7.2;2. Asymptotic Expansions;19
7.3;3. Generalized Asymptotic Expansions;25
7.4;4. Integration by Parts;29
7.5;5. Watson's Lemma;34
7.6;6. The Euler-Maclaurin Summation Formula;47
7.7;Exercises;57
7.8;Supplementary Notes;69
8;Chapter II. Classical Procedures;70
8.1;1. Laplace's Method;70
8.2;2. Logarithmic Singularities;81
8.3;3. The Principle of Stationary Phase;91
8.4;4. Method of Steepest Descents;99
8.5;5. Perron's Method;118
8.6;6. Darboux's Method;131
8.7;7. A Formula of Hayman;137
8.8;Exercises;143
8.9;Supplementary Notes;158
9;Chapter III. Mellin Transform Techniques;162
9.1;1. Introduction;162
9.2;2. Properties of Mellin Transforms;164
9.3;3. Examples;168
9.4;4. Work of Handelsman and Lew;172
9.5;5. Remarks and Examples;182
9.6;6. Explicit Error Terms;188
9.7;7. A Double Integral;191
9.8;Exercises;200
9.9;Supplementary Notes;206
9.10;SHORT TABLE OF MELLIN TRANSFORMS;207
10;Chapter IV. The Summability Method;210
10.1;1. Introduction;210
10.2;2. A Fourier Integral;213
10.3;3. Hankel Transform;218
10.4;4. Hankel Transform (continued);225
10.5;5. Oscillatory Kernels: General Case;231
10.6;6. Some Quadrature Formulas;236
10.7;7. Mellin-Barnes Type Integrals;243
10.8;Exercises;246
10.9;Supplementary Notes;255
11;Chapter V. Elementary Theory of
Distributions;256
11.1;1. Introduction;256
11.2;2. Test Functions and Distributions;257
11.3;3. Support of Distributions;259
11.4;4. Operations on Distributions;262
11.5;5. Differentiation of Distributions;264
11.6;6. Convolutions;269
11.7;7. Regularization of Divergent Integrals;272
11.8;8. Tempered Distributions;276
11.9;9. Distributions of Several Variables;280
11.10;10. The Distribution
r.;283
11.11;11. Taylor and Laurent Series for
r.;285
11.12;12. Fourier Transforms;289
11.13;13. Surface Distributions;295
11.14;Exercises;298
11.15;Supplementary Notes;305
12;Chapter VI. The Distributional
Approach;308
12.1;1. Introduction;308
12.2;2. The Stieltjes Transform;310
12.3;3. Stieltjes Transform: An Oscillatory Case;319
12.4;4. Hilbert Transforms;327
12.5;5. Laplace and Fourier Transforms Near the Origin;336
12.6;6. Fractional Integrals;341
12.7;7. The Method of Regularization;348
12.8;Exercises;361
12.9;Supplementary Notes;366
13;Chapter VII. Uniform Asymptotic
Expansions;368
13.1;1. Introduction;368
13.2;2. Saddle Point near a Pole;371
13.3;3. Saddle Point near an Endpoint;375
13.4;4. Two Coalescing Saddle Points;381
13.5;5. Laguerre Polynomials I;387
13.6;6. Many Coalescing Saddle Points;398
13.7;7. Laguerre Polynomials II;411
13.8;8. Legendre
Function;421
13.9;Exercises;424
13.10;Supplementary Notes;436
14;Chapter VIII. Double Integrals;438
14.1;1. Introduction;438
14.2;2. Classification of Critical Points;439
14.3;3. Local Extrema;443
14.4;4. Saddle Points;447
14.5;5. A Degenerate Case;450
14.6;6. Boundary Stationary Points;455
14.7;7. Critical Points of the Second Kind;457
14.8;8. Critical Points of the Third Kind;463
14.9;9. A Curve of Stationary Points;467
14.10;10. Laplace's Approximation;474
14.11;11. Boundary Extrema;482
14.12;Exercises;483
14.13;Supplementary Notes;490
15;Chapter IX. Higher Dimensional
Integrals;492
15.1;1. Introduction;492
15.2;2. Stationary Points;493
15.3;3. Points of Tangential Contact;502
15.4;4. Degenerate Stationary Point;506
15.5;5. Laplace's Approximation in
Rn;509
15.6;6. Multiple Fourier Transforms;515
15.7;Exercises;520
15.8;Supplementary Notes;529
16;Bibliography;532
17;Symbol Index;548
18;Author Index;550
19;Subject Index;554