Ramanujans Theta Functions

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Ramanujan's Theta Functions

Author : Shaun Cooper
Publisher : Springer
ISBN 13 : 3319561723
Total Pages : 687 pages
Book Rating : 4.3/5 (195 download)

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Book Synopsis Ramanujan's Theta Functions by : Shaun Cooper

Download or read book Ramanujan's Theta Functions written by Shaun Cooper and published by Springer. This book was released on 2017-06-12 with total page 687 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Ramanujan's Theta Functions

Author : Shaun Cooper
Publisher : Springer
ISBN 13 : 9783319858432
Total Pages : 687 pages
Book Rating : 4.8/5 (584 download)

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Book Synopsis Ramanujan's Theta Functions by : Shaun Cooper

Download or read book Ramanujan's Theta Functions written by Shaun Cooper and published by Springer. This book was released on 2018-08-02 with total page 687 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Ramanujan's Theta Functions

Author : Shaun Cooper
Publisher : Springer
ISBN 13 : 9783319561714
Total Pages : 0 pages
Book Rating : 4.5/5 (617 download)

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Book Synopsis Ramanujan's Theta Functions by : Shaun Cooper

Download or read book Ramanujan's Theta Functions written by Shaun Cooper and published by Springer. This book was released on 2017-06-26 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Ramanujan's Lost Notebook

Author : George E. Andrews
Publisher : Springer Science & Business Media
ISBN 13 : 9780387255293
Total Pages : 460 pages
Book Rating : 4.2/5 (552 download)

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Book Synopsis Ramanujan's Lost Notebook by : George E. Andrews

Download or read book Ramanujan's Lost Notebook written by George E. Andrews and published by Springer Science & Business Media. This book was released on 2005-05-06 with total page 460 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the library at Trinity College, Cambridge in 1976, George Andrews of Pennsylvania State University discovered a sheaf of pages in the handwriting of Srinivasa Ramanujan. Soon designated as "Ramanujan’s Lost Notebook," it contains considerable material on mock theta functions and undoubtedly dates from the last year of Ramanujan’s life. In this book, the notebook is presented with additional material and expert commentary.

Basic Hypergeometric Series and Applications

Author : Nathan Jacob Fine
Publisher : American Mathematical Soc.
ISBN 13 : 0821815245
Total Pages : 142 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Basic Hypergeometric Series and Applications by : Nathan Jacob Fine

Download or read book Basic Hypergeometric Series and Applications written by Nathan Jacob Fine and published by American Mathematical Soc.. This book was released on 1988 with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of partitions, founded by Euler, has led in a natural way to the idea of basic hypergeometric series, also known as Eulerian series. These series were first studied systematically by Heine, but many early results are attributed to Euler, Gauss, and Jacobi. This book provides a simple approach to basic hypergeometric series.

Theta Functions

Author : Maruti Ram Murty
Publisher : American Mathematical Soc.
ISBN 13 : 9780821870112
Total Pages : 188 pages
Book Rating : 4.8/5 (71 download)

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Book Synopsis Theta Functions by : Maruti Ram Murty

Download or read book Theta Functions written by Maruti Ram Murty and published by American Mathematical Soc.. This book was released on 1993-01-01 with total page 188 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains lectures on theta functions written by experts well known for excellence in exposition. The lectures represent the content of four courses given at the Centre de Recherches Mathematiques in Montreal during the academic year 1991-1992, which was devoted to the study of automorphic forms. Aimed at graduate students, the book synthesizes the classical and modern points of view in theta functions, concentrating on connections to number theory and representation theory. An excellent introduction to this important subject of current research, this book is suitable as a text in advanced graduate courses.

Number Theory in the Spirit of Ramanujan

Author : Bruce C. Berndt
Publisher : American Mathematical Soc.
ISBN 13 : 0821841785
Total Pages : 210 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Number Theory in the Spirit of Ramanujan by : Bruce C. Berndt

Download or read book Number Theory in the Spirit of Ramanujan written by Bruce C. Berndt and published by American Mathematical Soc.. This book was released on 2006 with total page 210 pages. Available in PDF, EPUB and Kindle. Book excerpt: Ramanujan is recognized as one of the great number theorists of the twentieth century. Here now is the first book to provide an introduction to his work in number theory. Most of Ramanujan's work in number theory arose out of $q$-series and theta functions. This book provides an introduction to these two important subjects and to some of the topics in number theory that are inextricably intertwined with them, including the theory of partitions, sums of squares and triangular numbers, and the Ramanujan tau function. The majority of the results discussed here are originally due to Ramanujan or were rediscovered by him. Ramanujan did not leave us proofs of the thousands of theorems he recorded in his notebooks, and so it cannot be claimed that many of the proofs given in this book are those found by Ramanujan. However, they are all in the spirit of his mathematics. The subjects examined in this book have a rich history dating back to Euler and Jacobi, and they continue to be focal points of contemporary mathematical research. Therefore, at the end of each of the seven chapters, Berndt discusses the results established in the chapter and places them in both historical and contemporary contexts. The book is suitable for advanced undergraduates and beginning graduate students interested in number theory.

Theta Functions, Elliptic Functions and [pi]

Author : Heng Huat Chan
Publisher : de Gruyter
ISBN 13 : 9783110540710
Total Pages : 0 pages
Book Rating : 4.5/5 (47 download)

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Book Synopsis Theta Functions, Elliptic Functions and [pi] by : Heng Huat Chan

Download or read book Theta Functions, Elliptic Functions and [pi] written by Heng Huat Chan and published by de Gruyter. This book was released on 2020 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

The Theory of Jacobi Forms

Author : Martin Eichler
Publisher : Springer Science & Business Media
ISBN 13 : 1468491628
Total Pages : 156 pages
Book Rating : 4.4/5 (684 download)

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Book Synopsis The Theory of Jacobi Forms by : Martin Eichler

Download or read book The Theory of Jacobi Forms written by Martin Eichler and published by Springer Science & Business Media. This book was released on 2013-12-14 with total page 156 pages. Available in PDF, EPUB and Kindle. Book excerpt: The functions studied in this monogra9h are a cross between elliptic functions and modular forms in one variable. Specifically, we define a Jacobi form on SL (~) to be a holomorphic function 2 (JC = upper half-plane) satisfying the t\-10 transformation eouations 2Tiimcz· k CT +d a-r +b z) (1) ((cT+d) e cp(T, z) cp CT +d ' CT +d (2) rjl(T, z+h+]l) and having a Four·ier expansion of the form 00 e2Tii(nT +rz) (3) cp(T, z) 2: c(n, r) 2:: rE~ n=O 2 r ~ 4nm Here k and m are natural numbers, called the weight and index of rp, respectively. Note that th e function cp (T, 0) is an ordinary modular formofweight k, whileforfixed T thefunction z-+rjl( -r, z) isa function of the type normally used to embed the elliptic curve ~/~T + ~ into a projective space. If m= 0, then cp is independent of z and the definition reduces to the usual notion of modular forms in one variable. We give three other examples of situations where functions satisfying (1)-(3) arise classically: 1. Theta series. Let Q: ~-+ ~ be a positive definite integer valued quadratic form and B the associated bilinear form.

George E. Andrews 80 Years of Combinatory Analysis

Author : Krishnaswami Alladi
Publisher : Birkhäuser
ISBN 13 : 9783030570521
Total Pages : 810 pages
Book Rating : 4.5/5 (75 download)

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Book Synopsis George E. Andrews 80 Years of Combinatory Analysis by : Krishnaswami Alladi

Download or read book George E. Andrews 80 Years of Combinatory Analysis written by Krishnaswami Alladi and published by Birkhäuser. This book was released on 2022-02-11 with total page 810 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a printed testimony for the fact that George Andrews, one of the world’s leading experts in partitions and q-series for the last several decades, has passed the milestone age of 80. To honor George Andrews on this occasion, the conference “Combinatory Analysis 2018” was organized at the Pennsylvania State University from June 21 to 24, 2018. This volume comprises the original articles from the Special Issue “Combinatory Analysis 2018 – In Honor of George Andrews’ 80th Birthday” resulting from the conference and published in Annals of Combinatorics. In addition to the 37 articles of the Andrews 80 Special Issue, the book includes two new papers. These research contributions explore new grounds and present new achievements, research trends, and problems in the area. The volume is complemented by three special personal contributions: “The Worlds of George Andrews, a daughter’s take” by Amy Alznauer, “My association and collaboration with George Andrews” by Krishna Alladi, and “Ramanujan, his Lost Notebook, its importance” by Bruce Berndt. Another aspect which gives this Andrews volume a truly unique character is the “Photos” collection. In addition to pictures taken at “Combinatory Analysis 2018”, the editors selected a variety of photos, many of them not available elsewhere: “Andrews in Austria”, “Andrews in China”, “Andrews in Florida”, “Andrews in Illinois”, and “Andrews in India”. This volume will be of interest to researchers, PhD students, and interested practitioners working in the area of Combinatory Analysis, q-Series, and related fields.

Partitions, q-Series, and Modular Forms

Author : Krishnaswami Alladi
Publisher : Springer Science & Business Media
ISBN 13 : 1461400287
Total Pages : 233 pages
Book Rating : 4.4/5 (614 download)

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Book Synopsis Partitions, q-Series, and Modular Forms by : Krishnaswami Alladi

Download or read book Partitions, q-Series, and Modular Forms written by Krishnaswami Alladi and published by Springer Science & Business Media. This book was released on 2011-11-01 with total page 233 pages. Available in PDF, EPUB and Kindle. Book excerpt: Partitions, q-Series, and Modular Forms contains a collection of research and survey papers that grew out of a Conference on Partitions, q-Series and Modular Forms at the University of Florida, Gainesville in March 2008. It will be of interest to researchers and graduate students that would like to learn of recent developments in the theory of q-series and modular and how it relates to number theory, combinatorics and special functions.

Chapter 16 of Ramanujan's Second Notebook: Theta-Functions and $q$-Series

Author : Chandrashekar Adiga
Publisher : American Mathematical Soc.
ISBN 13 : 0821823167
Total Pages : 99 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Chapter 16 of Ramanujan's Second Notebook: Theta-Functions and $q$-Series by : Chandrashekar Adiga

Download or read book Chapter 16 of Ramanujan's Second Notebook: Theta-Functions and $q$-Series written by Chandrashekar Adiga and published by American Mathematical Soc.. This book was released on 1985 with total page 99 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first part of Chapter 16 in Ramanujan's second notebook is devoted to q-series. Several of the results obtained by Ramanujan are classical, but many are new. In particular, certain elegant q-continued fraction expansions have not appeared heretofore in print. In the remainder of this chapter, Ramanujan develops the theory of the classical theta-functions in a manner different from his nineteenth century predecessors such as Jacobi. Although many of Ramanujan's discoveries about theta-functions are well-known, several new results are also to be found.

Surveys in Number Theory

Author : Krishnaswami Alladi
Publisher : Springer Science & Business Media
ISBN 13 : 0387785108
Total Pages : 193 pages
Book Rating : 4.3/5 (877 download)

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Book Synopsis Surveys in Number Theory by : Krishnaswami Alladi

Download or read book Surveys in Number Theory written by Krishnaswami Alladi and published by Springer Science & Business Media. This book was released on 2009-03-02 with total page 193 pages. Available in PDF, EPUB and Kindle. Book excerpt: Number theory has a wealth of long-standing problems, the study of which over the years has led to major developments in many areas of mathematics. This volume consists of seven significant chapters on number theory and related topics. Written by distinguished mathematicians, key topics focus on multipartitions, congruences and identities (G. Andrews), the formulas of Koshliakov and Guinand in Ramanujan's Lost Notebook (B. C. Berndt, Y. Lee, and J. Sohn), alternating sign matrices and the Weyl character formulas (D. M. Bressoud), theta functions in complex analysis (H. M. Farkas), representation functions in additive number theory (M. B. Nathanson), and mock theta functions, ranks, and Maass forms (K. Ono), and elliptic functions (M. Waldschmidt).

Ramanujan's Notebooks

Author : Srinivasa Ramanujan Aiyangar
Publisher :
ISBN 13 :
Total Pages : 536 pages
Book Rating : 4.:/5 (318 download)

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Book Synopsis Ramanujan's Notebooks by : Srinivasa Ramanujan Aiyangar

Download or read book Ramanujan's Notebooks written by Srinivasa Ramanujan Aiyangar and published by . This book was released on 1985 with total page 536 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Development of Elliptic Functions According to Ramanujan

Author : K. Venkatachaliengar
Publisher : World Scientific
ISBN 13 : 9814366455
Total Pages : 185 pages
Book Rating : 4.8/5 (143 download)

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Book Synopsis Development of Elliptic Functions According to Ramanujan by : K. Venkatachaliengar

Download or read book Development of Elliptic Functions According to Ramanujan written by K. Venkatachaliengar and published by World Scientific. This book was released on 2012 with total page 185 pages. Available in PDF, EPUB and Kindle. Book excerpt: This unique book provides an innovative and efficient approach to elliptic functions, based on the ideas of the great Indian mathematician Srinivasa Ramanujan. The original 1988 monograph of K Venkatachaliengar has been completely revised. Many details, omitted from the original version, have been included, and the book has been made comprehensive by notes at the end of each chapter. The book is for graduate students and researchers in Number Theory and Classical Analysis, as well for scholars and aficionados of Ramanujan's work. It can be read by anyone with some undergraduate knowledge of real and complex analysis.

Notebooks of Srinivasa Ramanujan

Author : Srinivasa Ramanujan Aiyangar
Publisher :
ISBN 13 :
Total Pages : 382 pages
Book Rating : 4.5/5 (32 download)

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Book Synopsis Notebooks of Srinivasa Ramanujan by : Srinivasa Ramanujan Aiyangar

Download or read book Notebooks of Srinivasa Ramanujan written by Srinivasa Ramanujan Aiyangar and published by . This book was released on 1957 with total page 382 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Harmonic Maass Forms and Mock Modular Forms: Theory and Applications

Author : Kathrin Bringmann
Publisher : American Mathematical Soc.
ISBN 13 : 1470419440
Total Pages : 391 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Harmonic Maass Forms and Mock Modular Forms: Theory and Applications by : Kathrin Bringmann

Download or read book Harmonic Maass Forms and Mock Modular Forms: Theory and Applications written by Kathrin Bringmann and published by American Mathematical Soc.. This book was released on 2017-12-15 with total page 391 pages. Available in PDF, EPUB and Kindle. Book excerpt: Modular forms and Jacobi forms play a central role in many areas of mathematics. Over the last 10–15 years, this theory has been extended to certain non-holomorphic functions, the so-called “harmonic Maass forms”. The first glimpses of this theory appeared in Ramanujan's enigmatic last letter to G. H. Hardy written from his deathbed. Ramanujan discovered functions he called “mock theta functions” which over eighty years later were recognized as pieces of harmonic Maass forms. This book contains the essential features of the theory of harmonic Maass forms and mock modular forms, together with a wide variety of applications to algebraic number theory, combinatorics, elliptic curves, mathematical physics, quantum modular forms, and representation theory.