Hadamard Expansions and Hyperasymptotic Evaluation: An Extension of the Method of Steepest Descents (Encyclopedia of Mathematics and its Applications)

The Author Describes The Recently Developed Theory Of Hadamard Expansions Applied To The High-precision (hyperasymptotic) Evaluation Of Laplace And Laplace-type Integrals. This Brand New Method Builds On The Well-known Asymptotic Method Of Steepest Descents, Of Which The Opening Chapter Gives A Detailed Account Illustrated By A Series Of Examples Of Increasing Complexity. A Discussion Of Uniformity Problems Associated With Various Coalescence Phenomena, The Stokes Phenomenon And Hyperasymptotics Of Laplace-type Integrals Follows. The Remaining Chapters Deal With The Hadamard Expansion Of Laplace Integrals, With And Without Saddle Points. Problems Of Different Types Of Saddle Coalescence Are Also Discussed. The Text Is Illustrated With Many Numerical Examples, Which Help The Reader To Understand The Level Of Accuracy Achievable. The Author Also Considers Applications To Some Important Special Functions. This Book Is Ideal For Graduate Students And Researchers Working In Asymptotics-- Machine Generated Contents Note: Preface; 1. Asymptotics Of Laplace-type Integrals; 2. Hadamard Expansion Of Laplace Integrals; 3. Hadamard Expansion Of Laplace-type Integrals; 4. Applications; Appendix A; Appendix B; Appendix C; References; Index. R.b. Paris. Includes Bibliographical References And Index. "The author describes the recently developed theory of Hadamard expansions applied to the high-precision (hyperasymptotic) evaluation of Laplace and Laplace-type integrals. This brand new method builds on the well-known asymptotic method of steepest descents, of which the opening chapter gives a detailed account illustrated by a series of examples of increasing complexity. A discussion of uniformity problems associated with various coalescence phenomena, the Stokes phenomenon and hyperasymptotics of Laplace-type integrals follows. The remaining chapters deal with the Hadamard expansion of Laplace integrals, with and without saddle points. Problems of different types of saddle coalescence are also discussed. The text is illustrated with many numerical examples, which help the reader to understand the level of accuracy achievable. The author also considers applications to some important special functions. This book is ideal for graduate students and researchers working in asymptotics"-- Provided by publisher For applied mathematicians and physical scientists interested in asymptotic calculations, this book describes a brand new method for the high-precision evaluation of Laplace integrals. Developed over the past decade, this method builds on the classical asymptotic method of steepest descents. Many numerical examples are included to illustrate the accuracy achievable.