Asymptotic expansions of Zeta functions. by Richard James Brown

Cover of: Asymptotic expansions of Zeta functions. | Richard James Brown

Published by University of Manchester in Manchester .

Written in English

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Edition Notes

Thesis (Ph.D.), - University of Manchester, Department of Mathematics.

Book details

ContributionsUniversity of Manchester. Department of Mathematics.
The Physical Object
Pagination231p.
Number of Pages231
ID Numbers
Open LibraryOL16564285M

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Request PDF | Asymptotic expansions of the Hurwitz–Lerch zeta function | The Hurwitz–Lerch zeta function Φ(z,s,a) is considered for large and small values of a∈C, and for large values of z. Asymptotic equalities for some weighted mean squares of R(a,x,s) are given in [6].

Integral representations as well as functional relations and expansions for Φ(z,s,a) may be found in [11, Section ]. Asymptotic expansions of R(a,x,s) have Cited by: Originally prepared for the Office of Naval Research, this important monograph introduces various methods for the asymptotic evaluation of integrals containing a large parameter, and solutions of ordinary linear differential equations by means of asymptotic expansions.

Asymptotic expansions An asymptotic expansion describes the asymptotic behavior of a function in terms of a sequence of gauge functions. The definition was introduced by Poincar´e (), and it provides a solid mathematical foundation for the use of many divergent series.

Definition A sequence of functions ϕFile Size: KB. Asymptotic expansions for a class of zeta-functions Article in The Ramanujan Journal 24(3) April with 15 Reads How we measure 'reads'. In this paper, we shall reveal the hidden structure in recent results of Katsurada as the Meijer G-function hierarchy.

In Sect. 1, we consider the holomorphic Eisenstein series and show that Katsurada’s two new expressions are variants of the classical Chowla–Selberg integral formula (Fourier expansion) with or without the beta-transform of Katsurada being Author: T. Kuzumaki. Asymptotic expansions of the double Zeta function.

A.C. KingUniform asymptotic expansions for the Barnes double gamma function. Proc. Roy. Soc. London Ser. A, (), pp. MatsumotoAsymptotic series for double zeta and double gamma functions of Barnes. RIMS Kokyuroku., (), pp.

Google ScholarCited by: 3. Asymptotic expansions of the Hurwitz-Lerch zeta function The integral representation (4) given above is the starting point to derive asymptotic expansions of Φ(z;s;a) Cited by:   Matsumoto, K.: Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series.

Nagoya Math. 59– () zbMATH MathSciNet Google Scholar Cited by: 7. Bessel functions, Hankel functions, asymptotic expansions for large argument, derivatives Notes: These results follow by differentiation of the corresponding expansions in §.

This book is the result of several years of work by the authors on different aspects of zeta functions and related topics. The aim is twofold. On one hand, a considerable number of useful formulas, essential for dealing with the different aspects of zeta-function regularization (analytic continuation, asymptotic expansions), many of which appear here, in book format, for the first.

Spectral zeta functions of graphs and the Riemann zeta [Lo12, Ly10, LPS14]. Terms appearing in our asymptotic expansions are zeta functions of lattice graphs and of continuous torus which are Epstein zeta function from number theory. This relies to an important extent on the work of Chinta, The Riemann zeta function is essentially the.

The Handbook of Special Functions provides in-depth coverage of special functions, which are used to help solve many of the most difficult problems in physics, engineering, and mathematics. The book presents new results along with well-known formulas used in many of the most important mathematical methods in order to solve a wide variety of.

'Asymptotics and Mellin-Barnes integrals by R. Paris and D. Kaminski is one of the first new, extended texts to be published in English since the recent advances began, and is a mixture of existing and novel techniques and applications.

but the comprehensive nature of this work means that it is likely to become one of the most significant textbook references for Mellin Cited by: Asymptotics and Special Functions book.

By Frank Olver. Edition 1st Edition. First Published The Zeta Function. View abstract. chapter 1 | 1 pages Integration by Parts. View abstract. Asymptotic Expansions by Laplace's Method; Gamma Function of Large Argument.

View abstract. chapter 9 Cited by: Comment Math Helvetici57 () 1 21 Birkhauser Verlag, Basel Zêta Functions and Their Asymptotic Expansions for Compact Symmetric Spaces of Rank One by Robert S Cahn (University of Miami, Coral Gables, Flonda) and Joseph A Wolf (University of Cahfornia, Berkeley, California) §0.

Introduction In this paper we apply E Cartan's theory of class 1. Whittaker Functions Examples of ${}_1F_1$ and Whittaker Functions Bessel's Equation and Bessel Functions Recurrence Relations Integral Representations of Bessel Functions Asymptotic Expansions Fourier Transforms and Bessel Functions Addition Theorems Integrals of Bessel Functions The Modified Bessel.

Properties of asymptotic expansions 26 Asymptotic expansions of integrals 29 Chapter 4. Laplace integrals 31 Laplace’s method 32 Watson’s lemma 36 Chapter 5.

Method of stationary phase 39 Chapter 6. Method of steepest descents 43 Bibliography 49 Appendix A. Notes 51 A Remainder theorem 51 A Taylor series for File Size: KB. You can write a book review and share your experiences. Other readers will always be interested in your opinion of the books you've read.

Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. Katsurada, MAsymptotic expansions for double Shintani zeta-functions of several variables. in Diophantine Analysis and Related FieldsDARF - AIP Conference Proceedings, vol.

pp.Diophantine Analysis and Related FieldsDARF -Musashino, Tokyo, Japan, 11/3/3. This book is the result of several years of work by the authors on different aspects of zeta functions and related topics.

The aim is twofold. On one hand, a considerable number of useful formulas, essential for dealing with the different aspects of zeta-function regularization (analytic continuation, asymptotic expansions), many of which.

Taylor Series and Asymptotic Expansions The importance of power series as a convenient representation, as an approximation tool, as a tool for solving differential equations and so on, is pretty obvious.

What may not be so obvious is that power series can be of some use even when they diverge. Let us start by considering Taylor Size: KB. Earlier investigations on uniform asymptotic expansions of the incomplete gamma functions are reconsidered.

The new results include estimations for the remainder and the extension of the results to complex by: Asymptotic Expansions: Their Derivation and Interpretation by R B Dingle I have decided to host Dingle’s book on my home page because it is long out of print and much in demand.

He describes how divergent series originate, how their terms can be calculated, and above all how they can be regarded as exact coded. asymptotic expansions.

We also obtain there the asymptotic expansion of the recently introduced Arakawa-Kaneko zeta function (Example 7 in Section 5). Most remarkably, the poly-Bernoulli polynomials and the Bernoulli polynomials appear in several important asymptotic expansions for a large class of functions – see Corollary 2 in Section 2.

ASYMPTOTIC PROPERTIES OF ZETA FUNCTIONS OVER FINITE FIELDS ALEXEY ZYKIN Abstract. In this paper we study asymptotic properties of families of zeta and L-functions over nite elds.

We do it in the context of three main prob-lems: the basic inequality, the Brauer{Siegel type results and the results on distribution of zeroes.

Dynamical zeta functions and asymptotic expansions in Nielsen theory 67 78; Computing the Riemann zeta function by numerical quadrature 81 92; On Riemann's zeta function 93 ; A prime orbit theorem for self-similar flows and Diophantine approximation ; The nd zero of the Riemann zeta function Bibliography.

Zeta and q-Zeta Functions and Associated Series and Integrals, Cited by: 1. Asymptotic series for the Hurwitz zeta function, its derivative, and related functions (including the Riemann zeta function of odd integer argument) are derived as an illustration of a simple, direct method of broad applicability, inspired by the calculus of finite by: 5.

This book details the classical part of the theory of algebraic number theory, excluding class-field theory and its consequences. Coverage includes: ideal theory in rings of algebraic integers, p-adic fields and their finite extensions, ideles and adeles, zeta-functions, distribution of prime ideals, Abelian fields, the class-number of quadratic fields, and factorization problems.

Topics include Bessel functions of zero order, modified Bessel functions, definite integrals, asymptotic expansions, and Bessel functions of any real order.

More than problems throughout the book enable students to test and extend their understanding of the theory and applications of Bessel by: The problem of function asymptotic expansion in prescribed system of functions requires information about class of function [7].

For the function defined by power series this problem can be solved with the aid of mentioned methods by expansion of approximant resulted.

This way is concerned with sequence of intermediateAuthor: Mihail Nikitin. explicit formulas and asymptotic expansions for certain mean square of hurwitz zeta-functions. In Analytic and Probabilistic Methods in Number Theory: Proceedings of the Second International Conference in Honour of J.

Kubilius, Palanga, Lithuania, September (pp. Asymptotic expansions, derivation and interpretation R.B. Dingle. Categories: Mathematics\\Analysis. Year: Other readers will always be interested in your opinion of the books you've read. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them.

In mathematics, an Igusa zeta function is a type of generating function, counting the number of solutions of an equation, modulo p, p 2, p 3, and so on. Definition. For a Igusa, Jun-Ichi (), "Complex powers and asymptotic expansions.

Functions of certain types", Journal für die reine und angewandte Mathematik,   The book concludes with an evaluation of methods used in estimating (as opposed to bounding) errors in asymptotic approximations and expansions.

This monograph is intended for graduate mathematicians, physicists, and Edition: 1. This book is the result of several years of work by the authors on different aspects of zeta functions and related topics.

The aim is twofold. On one hand, a considerable number of useful formulas, essential for dealing with the different aspects of zeta-function regularization (analytic continuation, asymptotic expansions), many of which Cited by: ctional_merge_sorted (A, B, key=None) Merge the two tuples/lists, keeping the orders provided by them.

INPUT: A – a list or tuple (type has to coincide with type of B).; B – a list or tuple (type has to coincide with type of A).; key – (default: None) a None, then the identity is key-function applied on an element of the list.

The large parameter expansion plays a determining role in the behavior of these Bose systems in the limit that the external magnetic field B approaches 0. This particular expansion is generalized herein and its validity tested by determining the asymptotic. Composite Asymptotic Expansions. by Augustin Fruchard,Reinhard Schafke.

Lecture Notes in Mathematics (Book ) Thanks for Sharing. You submitted the following rating and review. We'll publish them on our site once we've reviewed : Springer Berlin Heidelberg. DynamicaL Spectral, and Arithmetic Zeta Functions JanuarySan Antonio, Texas Dynamical zeta functions and asymptotic expansions in Nielsen theory Spectral and Arithmetic Zeta Functions', held at the Annual Meeting of the American Mathe.Finding asymptotic expansion.

Ask Question Asked 4 years, 6 months ago. Active 4 years, Multiplication of two asymptotic expansions. 2. Asymptotic Expansion of an Integral involving Modified Bessel Functions. 6.Asymptotic expansion of Ai and Bi.

The Stokes phenomenon. Properties of Airy functions. Zeros of Airy functions. The spectral zeta function. Inequalities. Connection with Bessel functions. Modulus and phase of Airy functions.

Definitions. Differential equations. Asymptotic expansions. Functions of positive arguments. Inhomogeneous Airy.

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