Theta Functions Elliptic Functions And Pi

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Theta Functions Elliptic Functions And

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Author : Heng Huat Chan
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2020-07-06

Theta Functions Elliptic Functions And written by Heng Huat Chan and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-07-06 with Mathematics categories.

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.

Theta Functions Elliptic Functions And Pi

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Author : Heng Huat Chan
language : en
Publisher: de Gruyter
Release Date : 2020

Theta Functions Elliptic Functions And Pi written by Heng Huat Chan and has been published by de Gruyter this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020 with Elliptic functions categories.

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.

Elliptic Functions And Iterative Algorithms For Pi

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Author : Eduardo Evans
language : en
Publisher:
Release Date : 2023

Elliptic Functions And Iterative Algorithms For Pi written by Eduardo Evans and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023 with Elliptic functions -- Testing categories.

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.

A Brief Introduction To Theta Functions

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Author : Richard Bellman
language : en
Publisher: Courier Corporation
Release Date : 2013-01-01

A Brief Introduction To Theta Functions written by Richard Bellman and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-01-01 with Mathematics categories.

Originally published: New York: Rinehart and Winston, 1961.

Tables Of Theta Functions Elliptic Integrals K And E And Associated Coefficients In The Numerical Calculati On Of Elliptic Functions

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Author : Hantaro Nagaoka
language : en
Publisher:
Release Date : 1922

Tables Of Theta Functions Elliptic Integrals K And E And Associated Coefficients In The Numerical Calculati On Of Elliptic Functions written by Hantaro Nagaoka and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1922 with Elliptic functions categories.

Elliptic And Theta Functions

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Author : Eric Temple Bell
language : en
Publisher:
Release Date : 1925

Elliptic And Theta Functions written by Eric Temple Bell and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1925 with Elliptic functions categories.

Ten Place Tables Of The Jacobian Elliptic Functions

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Author : Henry E. Fettis
language : en
Publisher:
Release Date : 1966

Ten Place Tables Of The Jacobian Elliptic Functions written by Henry E. Fettis and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1966 with Elliptic functions categories.

Theta Functions

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Author : Jun-ichi Igusa
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Theta Functions written by Jun-ichi Igusa and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

The theory of theta functions has a long history; for this, we refer A. Krazer and W. Wirtinger the reader to an encyclopedia article by ("Sources" [9]). We shall restrict ourselves to postwar, i. e., after 1945, periods. Around 1948/49, F. Conforto, c. L. Siegel, A. Well reconsidered the main existence theorems of theta functions and found natural proofs for them. These are contained in Conforto: Abelsche Funktionen und algebraische Geometrie, Springer (1956); Siegel: Analytic functions of several complex variables, Lect. Notes, I.A.S. (1948/49); Well: Theoremes fondamentaux de la theorie des fonctions theta, Sem. Bourbaki, No. 16 (1949). The complete account of Weil's method appeared in his book of 1958 [20]. The next important achievement was the theory of compacti fication of the quotient variety of Siegel's upper-half space by a modular group. There are many ways to compactify the quotient variety; we are talking about what might be called a standard compactification. Such a compactification was obtained first as a Hausdorff space by I. Satake in "On the compactification of the Siegel space", J. Ind. Math. Soc. 20, 259-281 (1956), and as a normal projective variety by W.L. Baily in 1958 [1]. In 1957/58, H. Cartan took up this theory in his seminar [3]; it was shown that the graded ring of modular forms relative to the given modular group is a normal integral domain which is finitely generated over C

Ramanujan S Theta Functions

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Author : Shaun Cooper
language : en
Publisher: Springer
Release Date : 2017-06-12

Ramanujan S Theta Functions written by Shaun Cooper and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-06-12 with Mathematics categories.

Theta functions were studied extensively by Ramanujan. This book provides a systematic development of Ramanujan’s results and extends them to a general theory. The author’s treatment of the subject is comprehensive, providing a detailed study of theta functions and modular forms for levels up to 12. Aimed at advanced undergraduates, graduate students, and researchers, the organization, user-friendly presentation, and rich source of examples, lends this book to serve as a useful reference, a pedagogical tool, and a stimulus for further research. Topics, especially those discussed in the second half of the book, have been the subject of much recent research; many of which are appearing in book form for the first time. Further results are summarized in the numerous exercises at the end of each chapter.

Elliptic Functions And Elliptic Integrals

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Author : Viktor Vasil_evich Prasolov
language : en
Publisher: American Mathematical Soc.
Release Date : 1997-09-16

Elliptic Functions And Elliptic Integrals written by Viktor Vasil_evich Prasolov and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997-09-16 with Mathematics categories.

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.