## A Distributional Approach to Asymptotics: Theory and ApplicationsKey features of this significantly expanded second edition: - addition of several new chapters and sections, including a presentation of time-domain asymptotics needed for the understanding of wavelet theory - extensive examples and problem sets - useful bibliography and index. This book is a modern introduction to asymptotic analysis intended not only for mathematicians, but for physicists, engineers, and graduate students as well. Written by two of the leading experts in the field, the text provides readers with a firm grasp of mathematical theory, and at the same time demonstrates applications in areas such as differential equations, quantum mechanics, noncommutative geometry, and number theory. "...The authors of this remarkable book are among the very few who have faced up to the challenge of explaining what an asymptotic expansion is, and of systematizing the handling of asymptotic series. The idea of using distributions is an original one, and we recommend that you read the book...[it] should be on your bookshelf if you are at all interested in knowing what an asymptotic series is." ¿ "The Bulletin of Mathematics Books" (Review of the 1st edition) "...The book is a valuable one, one that many applied mathematicians may want to buy. The authors are undeniably experts in their field...most of the material has appeared in no other book." ¿ "SIAM News" (Review of the 1st edition) Table of contents Preface 1. Basic Results in Asymptotics 2. Introduction to the Theory of Distributions 3. A Distributional Theory for Asymptotic Expansions 4. The Asymptotic Expansion of Multi-Dimensional Generalized Functions 5. The Asymptotic Expansion of Certain Series Considered by Ramamujan 6. The Cesaro Behavior of Distributions 7. Series of Dirac Delta Functions References Index |

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### Contents

Baslc Results ln Asymptotlcs | 1 |

Introductlon to the Theory of Distributions | 53 |

A Distributional Theory for Asymptotic Expansions | 107 |

Asymptotic Expansion of Multidimensional Generalized Functions | 187 |

Asymptotic Expansion of Certain Series Considered | 249 |

Contents ix | 297 |

Series of Dlrac Delta Functions | 397 |

431 | |

447 | |

### Common terms and phrases

Abel summability admits an analytic analytic continuation argz associated homogeneous asymptotic development asymptotic sequence belongs boundary Cesaro sense Cesaro summable change of variables Chapter coefficients compact constant continuous function convergence critical points definition derivative differential Dirac delta functions distributionally small sequence divergent equation evaluation Example exists f(kx f(Xx Find the asymptotic finite follows Fourier series Fourier transform function defined function f(x given homogeneous functions homogeneous of degree infinity kernel Laplace's formula leading term Lemma Let f e Let us consider locally integrable function moments notation Observe operator pointwise polynomial primitive of order principal value Proof Prove Radon measure regularly varying result satisfies Section seminorms Show smooth function solution space spectral density supp support bounded Suppose Taylor series tempered distribution test function Theorem topology vanish yields