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Author: Heng Huat Chan
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110541912
Category : Mathematics
Languages : en
Pages : 138
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Book Description
This book presents several results on elliptic functions and Pi, using Jacobi’s triple product identity as a tool to show suprising connections between different topics within number theory such as theta functions, Eisenstein series, the Dedekind delta function, and Ramanujan’s work on Pi. The included exercises make it ideal for both classroom use and self-study.
Author: Heng Huat Chan
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110541912
Category : Mathematics
Languages : en
Pages : 138
Get Book Here
Book Description
This book presents several results on elliptic functions and Pi, using Jacobi’s triple product identity as a tool to show suprising connections between different topics within number theory such as theta functions, Eisenstein series, the Dedekind delta function, and Ramanujan’s work on Pi. The included exercises make it ideal for both classroom use and self-study.
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Author: Heng Huat Chan
Publisher: de Gruyter
ISBN: 9783110540710
Category : Elliptic functions
Languages : en
Pages : 0
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Book Description
This book presents several results on elliptic functions and Pi, using Jacobi's triple product identity as a tool to show suprising connections between different topics within number theory such as theta functions, Eisenstein series, the Dedekind delta function, and Ramanujan's work on Pi. The included exercises make it ideal for both classroom use and self-study.
Author: Hantaro Nagaoka
Publisher:
ISBN:
Category : Elliptic functions
Languages : en
Pages : 78
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Book Description
Author: Richard Bellman
Publisher: Courier Corporation
ISBN: 0486492958
Category : Mathematics
Languages : en
Pages : 100
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Book Description
Originally published: New York: Rinehart and Winston, 1961.
Author: Eduardo Evans
Publisher:
ISBN:
Category : Elliptic functions -- Testing
Languages : en
Pages : 0
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Book Description
Preliminary identities in the theory of basic hypergeometric series, or `q-series', are proven. These include q-analogues of the exponential function, which lead to a fairly simple proof of Jacobi's celebrated triple product identity due to Andrews. The Dedekind eta function is introduced and a few identities of it derived. Euler's pentagonal number theorem is shown as a special case of Ramanujan's theta function and Watson's quintuple product identity is proved in a manner given by Carlitz and Subbarao. The Jacobian theta functions are introduced as special kinds of basic hypergeometric series and various relations between them derived using the triple product identity, among other previously established results. A special quotient of theta functions is introduced as the modular lambda function. The Eisenstein series are first defined through their Lambert series expansions and a series of differential equations due to Ramanujan are developed. Modular forms and functions and subsequently elliptic functions are introduced. The Weierstrass p-function is developed along other elliptic functions, those being defined as certain quotients of theta functions. The first few Eisenstein series are then shown to be expressible in terms of theta functions. Theta functions are shown to be related to Gauss' hypergeometric series _2F_1(a,b;c;z) through the Jacobi inversion theorem. This is shown to have use in relating modular equations and hypergeometric series to pi. The arithmetic-geometric mean iteration of Gauss is developed and used in conjunction with other results established in proofs of two iterative algorithms for pi. Recent applications of pi algorithms using and not using the techniques developed here are then discussed.
Author: Henry E. Fettis
Publisher:
ISBN:
Category : Elliptic functions
Languages : en
Pages : 136
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Book Description
Author: Viktor Vasil_evich Prasolov
Publisher: American Mathematical Soc.
ISBN: 9780821897805
Category : Mathematics
Languages : en
Pages : 202
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Book Description
This book is devoted to the geometry and arithmetic of elliptic curves and to elliptic functions with applications to algebra and number theory. It includes modern interpretations of some famous classical algebraic theorems such as Abel's theorem on the lemniscate and Hermite's solution of the fifth degree equation by means of theta functions. Suitable as a text, the book is self-contained and assumes as prerequisites only the standard one-year courses of algebra and analysis.
Author: Eric Temple Bell
Publisher:
ISBN:
Category : Elliptic functions
Languages : en
Pages : 338
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Book Description
Author: Andre Weil
Publisher: Springer Science & Business Media
ISBN: 9783540650362
Category : Mathematics
Languages : en
Pages : 112
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Book Description
Drawn from the Foreword: (...) On the other hand, since much of the material in this volume seems suitable for inclusion in elementary courses, it may not be superfluous to point out that it is almost entirely self-contained. Even the basic facts about trigonometric functions are treated ab initio in Ch. II, according to Eisenstein's method. It would have been both logical and convenient to treat the gamma -function similarly in Ch. VII; for the sake of brevity, this has not been done, and a knowledge of some elementary properties of T(s) has been assumed. One further prerequisite in Part II is Dirichlet's theorem on Fourier series, together with the method of Poisson summation which is only a special case of that theorem; in the case under consideration (essentially no more than the transformation formula for the theta-function) this presupposes the calculation of some classical integrals. (...) As to the final chapter, it concerns applications to number theory (...).
Author: David Mumford
Publisher: Springer Science & Business Media
ISBN: 0817645772
Category : Mathematics
Languages : en
Pages : 248
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Book Description
This volume is the first of three in a series surveying the theory of theta functions. Based on lectures given by the author at the Tata Institute of Fundamental Research in Bombay, these volumes constitute a systematic exposition of theta functions, beginning with their historical roots as analytic functions in one variable (Volume I), touching on some of the beautiful ways they can be used to describe moduli spaces (Volume II), and culminating in a methodical comparison of theta functions in analysis, algebraic geometry, and representation theory (Volume III).
