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Hadamard Expansions and Hyperasymptotic Evaluation: An Extension of the Method of Steepest Descents (Encyclopedia of Mathematics,Used
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The author describes the recently developed theory of Hadamard expansions applied to the highprecision (hyperasymptotic) evaluation of Laplace and Laplacetype integrals. This brand new method builds on the wellknown 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 Laplacetype 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.
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