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Book

Asymptotic Approximations of Integrals

29 Sep 2014-
TL;DR: The basic concepts of asymptotic expansions, Mellin transform techniques, and the distributional approach are explained.
Abstract: Preface 1. Fundamental concepts of asymptotics 2. Classical procedures 3. Mellin transform techniques 4. The summability method 5. Elementary theory of distributions 6. The distributional approach 7. Uniform asymptotic expansions 8. Double integrals 9. Higher dimensional integrals Bibliography Symbol Index Author index Subject index.
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Book
01 Jan 2009
TL;DR: This text can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study, and is certain to become the definitive reference on the topic.
Abstract: Analytic Combinatorics is a self-contained treatment of the mathematics underlying the analysis of discrete structures, which has emerged over the past several decades as an essential tool in the understanding of properties of computer programs and scientific models with applications in physics, biology and chemistry. Thorough treatment of a large number of classical applications is an essential aspect of the presentation. Written by the leaders in the field of analytic combinatorics, this text is certain to become the definitive reference on the topic. The text is complemented with exercises, examples, appendices and notes to aid understanding therefore, it can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study.

3,616 citations


Cites background or methods from "Asymptotic Approximations of Integr..."

  • ...Because of its roots in app lied mathematics, the method is well covered by the literature in this area, and we refer to the book s by Olver [465], Henrici [329], or Wong [614] for extensive discussions....

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  • ...Mellin analysis of “harmonic integrals” is a classical topic of applied mathem atics for which we refer to the books by Wong [614] and Paris–Kaminski [472]....

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  • ..., the presen tation in [75, 465, 614]) and the already evoked r ôle of the Airy function....

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  • ...Hayman’s approach [325] which we have expounded here (see also [614]) is notable in its generality as it envisions saddle-point analysis in an abstract perspecti ve, which makes it possible to develop general closure theorems....

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  • ...The proof then proceeds with the analysis of the polylogarit hm whenz= ei (w−π) and s = 1/2+ i t , the integral (48) being estimated asymptotically as a harmonic integral (a continuous analogue of harmonic sums [614]) by means of Me llin transforms....

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Journal ArticleDOI
TL;DR: This survey presents a unified and essentially self-contained approach to the asymptotic analysis of a large class of sums that arise in combinatorial mathematics, discrete probabilistic models, and the average-case analysis of algorithms using the Mellin transform, a close relative of the integral transforms of Laplace and Fourier.

603 citations

Book
01 Mar 1996
TL;DR: In this article, the saddle point and large singularities of analytic functions are used for graphical enumeration, including implicit functions, recurrences, and combinations of methods, which are related to our work.
Abstract: 12 Large singularities of analytic functions 113 12.1 The saddle point 13 Multivariate generating functions 128 14 Mellin and other integral transforms 134 15 Functional equations, recurrences, and combinations of methods 137 15.1 Implicit functions, graphical enumeration, and related

385 citations

Book
18 Dec 2008
TL;DR: Markov Bases and Likelihood Inference have been used in this article for conditional independence and conditional independence in the context of Bayesian Integrals, and they have been shown to work well with open problems.
Abstract: Markov Bases.- Likelihood Inference.- Conditional Independence.- Hidden Variables.- Bayesian Integrals.- Exercises.- Open Problems.

368 citations

Journal ArticleDOI
TL;DR: In this article, the authors investigated the nonequilibrium evolution of the block entanglement entropy in a transverse magnetic field after the Hamiltonian parameters are suddenly changed from and to arbitrary values.
Abstract: The nonequilibrium evolution of the block entanglement entropy is investigated in the $XY$ chain in a transverse magnetic field after the Hamiltonian parameters are suddenly changed from and to arbitrary values. Using Toeplitz matrix representation and multidimensional phase methods, we provide analytic results for large blocks and for all times, showing explicitly the linear growth in time followed by saturation. The consequences of these analytic results are discussed and the effects of a finite block length is taken into account numerically.

313 citations


Additional excerpts

  • ...[14]), we consider the Toeplitz and the Hankel symbols Tkj = ∫ π...

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