Asymptotic Approximations Of Integrals
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Asymptotic Approximations Of Integrals
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Author : R. Wong
language : en
Publisher: Academic Press
Release Date : 2014-05-10
Asymptotic Approximations Of Integrals written by R. Wong and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-05-10 with Mathematics categories.
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.
Asymptotic Approximations Of Integrals
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Author :
language : en
Publisher:
Release Date : 2001
Asymptotic Approximations Of Integrals written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with categories.
Asymptotic Approximations Of Integrals
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Author : Roderick Wong
language : en
Publisher: Boston [Mass.] ; Toronto : Academic Press
Release Date : 1989
Asymptotic Approximations Of Integrals written by Roderick Wong and has been published by Boston [Mass.] ; Toronto : Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1989 with Mathematics categories.
Asymptotic Methods For Integrals
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Author : Nico M. Temme
language : en
Publisher: World Scientific Publishing Company
Release Date : 2015
Asymptotic Methods For Integrals written by Nico M. Temme and has been published by World Scientific Publishing Company this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015 with Differential equations categories.
This book gives introductory chapters on the classical basic and standard methods for asymptotic analysis, such as Watson's lemma, Laplace's method, the saddle point and steepest descent methods, stationary phase and Darboux's method. The methods, explained in great detail, will obtain asymptotic approximations of the well-known special functions of mathematical physics and probability theory. After these introductory chapters, the methods of uniform asymptotic analysis are described in which several parameters have influence on typical phenomena: turning points and transition points, coinciding saddle and singularities. In all these examples, the special functions are indicated that describe the peculiar behavior of the integrals. The text extensively covers the classical methods with an emphasis on how to obtain expansions, and how to use the results for numerical methods, in particular for approximating special functions. In this way, we work with a computational mind: how can we use certain expansions in numerical analysis and in computer programs, how can we compute coefficients, and so on.
Asymptotic Approximations For Probability Integrals
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Author : Karl W. Breitung
language : en
Publisher: Springer
Release Date : 2006-11-14
Asymptotic Approximations For Probability Integrals written by Karl W. Breitung and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-14 with Technology & Engineering categories.
This book gives a self-contained introduction to the subject of asymptotic approximation for multivariate integrals for both mathematicians and applied scientists. A collection of results of the Laplace methods is given. Such methods are useful for example in reliability, statistics, theoretical physics and information theory. An important special case is the approximation of multidimensional normal integrals. Here the relation between the differential geometry of the boundary of the integration domain and the asymptotic probability content is derived. One of the most important applications of these methods is in structural reliability. Engineers working in this field will find here a complete outline of asymptotic approximation methods for failure probability integrals.
Asymptotic Approximations Of Integrals With Applications
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Author :
language : en
Publisher:
Release Date : 2020
Asymptotic Approximations Of Integrals With Applications written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020 with Electronic books categories.
This thesis analyzes asymptotic approximations expansions for integrals, with several examples given. The main part of this research consists in studying the function g(x) = (1 + 1=x)x, which has the limit e as x → ∞. In a 2014 paper C.-P. Chen and J. Choi previously studied this function through an asymptotic expansion for large x. Our main focus is on the coefficients that appear in their expansion. Chen and Choi obtained an explicit formula for the coefficients, but it involves a sum of terms that grows exponentially in number. Our contribution is to find the coefficients in a more practical way, and also to determine the asymptotic behavior of the nth term as n → ∞. We derive a recursion formula, and we show it is simple to use and is numerically stable. We then use Cauchy’s integral formula to derive an explicit integral representation for the coefficients. From this we approximate the late coefficients by residue theory, and this approximation consists of two simple terms. We show the accuracy of the approximation with some numerical examples. We finally determine an integral representation of the error term in our asymptotic approximation, and from this show that is of smaller order of magnitude than the two leading terms.
Asymptotic Expansions Of Integrals
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Author : Norman Bleistein
language : en
Publisher: Courier Corporation
Release Date : 1986-01-01
Asymptotic Expansions Of Integrals written by Norman Bleistein and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 1986-01-01 with Mathematics categories.
Excellent introductory text, written by two experts, presents a coherent and systematic view of principles and methods. Topics include integration by parts, Watson's lemma, LaPlace's method, stationary phase, and steepest descents. Additional subjects include the Mellin transform method and less elementary aspects of the method of steepest descents. 1975 edition.
Computing Highly Oscillatory Integrals
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Author : Alfredo Deano
language : en
Publisher: SIAM
Release Date : 2018-01-01
Computing Highly Oscillatory Integrals written by Alfredo Deano and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-01-01 with Mathematics categories.
Highly oscillatory phenomena range across numerous areas in science and engineering and their computation represents a difficult challenge. A case in point is integrals of rapidly oscillating functions in one or more variables. The quadrature of such integrals has been historically considered very demanding. Research in the past 15 years (in which the authors played a major role) resulted in a range of very effective and affordable algorithms for highly oscillatory quadrature. This is the only monograph bringing together the new body of ideas in this area in its entirety. The starting point is that approximations need to be analyzed using asymptotic methods rather than by more standard polynomial expansions. As often happens in computational mathematics, once a phenomenon is understood from a mathematical standpoint, effective algorithms follow. As reviewed in this monograph, we now have at our disposal a number of very effective quadrature methods for highly oscillatory integrals--Filon-type and Levin-type methods, methods based on steepest descent, and complex-valued Gaussian quadrature. Their understanding calls for a fairly varied mathematical toolbox--from classical numerical analysis, approximation theory, and theory of orthogonal polynomials all the way to asymptotic analysis--yet this understanding is the cornerstone of efficient algorithms.
Asymptotics And Mellin Barnes Integrals
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Author : R. B. Paris
language : en
Publisher: Cambridge University Press
Release Date : 2001-09-24
Asymptotics And Mellin Barnes Integrals written by R. B. Paris and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001-09-24 with Mathematics categories.
Asymptotics and Mellin-Barnes Integrals provides an account of the use and properties of a type of complex integral representation that arises frequently in the study of special functions typically of interest in classical analysis and mathematical physics. After developing the properties of these integrals, their use in determining the asymptotic behavior of special functions is detailed. Although such integrals have a long history, the book's account includes recent research results in analytic number theory and hyperasymptotics. The book also fills a gap in the literature on asymptotic analysis and special functions by providing a thorough account of the use of Mellin-Barnes integrals that is otherwise not available in standard references on asymptotics.
Approximating Integrals Via Monte Carlo And Deterministic Methods
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Author : Michael Evans
language : en
Publisher: OUP Oxford
Release Date : 2000-03-23
Approximating Integrals Via Monte Carlo And Deterministic Methods written by Michael Evans and has been published by OUP Oxford this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000-03-23 with Mathematics categories.
This book is designed to introduce graduate students and researchers to the primary methods useful for approximating integrals. The emphasis is on those methods that have been found to be of practical use, and although the focus is on approximating higher- dimensional integrals the lower-dimensional case is also covered. Included in the book are asymptotic techniques, multiple quadrature and quasi-random techniques as well as a complete development of Monte Carlo algorithms. For the Monte Carlo section importance sampling methods, variance reduction techniques and the primary Markov Chain Monte Carlo algorithms are covered. This book brings these various techniques together for the first time, and hence provides an accessible textbook and reference for researchers in a wide variety of disciplines.