Ramanujan's Theta Functions

Ramanujan's Theta Functions
Author: Shaun Cooper
Publisher: Springer
Total Pages: 696
Release: 2017-06-12
Genre: Mathematics
ISBN: 3319561723

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.

Number Theory in the Spirit of Ramanujan

Number Theory in the Spirit of Ramanujan
Author: Bruce C. Berndt
Publisher: American Mathematical Soc.
Total Pages: 210
Release: 2006
Genre: Mathematics
ISBN: 0821841785

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.

A Course in Analysis

A Course in Analysis
Author: Niels Jacob
Publisher: World Scientific Publishing Company
Total Pages: 0
Release: 2016
Genre: Calculus
ISBN: 9789814689090

This volume covers the contents of two typical modules in an undergraduate mathematics course: part 1 - introductory calculus and part 2 - analysis of functions of one variable. The book contains 360 problems with complete solutions

Theta Functions, Bowdoin 1987

Theta Functions, Bowdoin 1987
Author: Leon Ehrenpreis
Publisher: American Mathematical Soc.
Total Pages: 730
Release: 1989
Genre: Mathematics
ISBN: 0821814834

During his long and productive career, Salomon Bochner worked in a variety of different areas of mathematics. This four part set brings together his collected papers, illustrating the range and depth of his mathematical interests. The books are available either individually or as a set.

My Search for Ramanujan

My Search for Ramanujan
Author: Ken Ono
Publisher: Springer
Total Pages: 235
Release: 2016-04-20
Genre: Mathematics
ISBN: 3319255681

"The son of a prominent Japanese mathematician who came to the United States after World War II, Ken Ono was raised on a diet of high expectations and little praise. Rebelling against his pressure-cooker of a life, Ken determined to drop out of high school to follow his own path. To obtain his father’s approval, he invoked the biography of the famous Indian mathematical prodigy Srinivasa Ramanujan, whom his father revered, who had twice flunked out of college because of his single-minded devotion to mathematics. Ono describes his rocky path through college and graduate school, interweaving Ramanujan’s story with his own and telling how at key moments, he was inspired by Ramanujan and guided by mentors who encouraged him to pursue his interest in exploring Ramanujan’s mathematical legacy. Picking up where others left off, beginning with the great English mathematician G.H. Hardy, who brought Ramanujan to Cambridge in 1914, Ono has devoted his mathematical career to understanding how in his short life, Ramanujan was able to discover so many deep mathematical truths, which Ramanujan believed had been sent to him as visions from a Hindu goddess. And it was Ramanujan who was ultimately the source of reconciliation between Ono and his parents. Ono’s search for Ramanujan ranges over three continents and crosses paths with mathematicians whose lives span the globe and the entire twentieth century and beyond. Along the way, Ken made many fascinating discoveries. The most important and surprising one of all was his own humanity."

Number Theory and Modular Forms

Number Theory and Modular Forms
Author: Bruce C. Berndt
Publisher: Springer Science & Business Media
Total Pages: 392
Release: 2013-11-11
Genre: Mathematics
ISBN: 1475760442

Robert A. Rankin, one of the world's foremost authorities on modular forms and a founding editor of The Ramanujan Journal, died on January 27, 2001, at the age of 85. Rankin had broad interests and contributed fundamental papers in a wide variety of areas within number theory, geometry, analysis, and algebra. To commemorate Rankin's life and work, the editors have collected together 25 papers by several eminent mathematicians reflecting Rankin's extensive range of interests within number theory. Many of these papers reflect Rankin's primary focus in modular forms. It is the editors' fervent hope that mathematicians will be stimulated by these papers and gain a greater appreciation for Rankin's contributions to mathematics. This volume would be an inspiration to students and researchers in the areas of number theory and modular forms.

Ramanujan’s Notebooks

Ramanujan’s Notebooks
Author: Bruce C. Berndt
Publisher: Springer Science & Business Media
Total Pages: 630
Release: 2012-12-06
Genre: Mathematics
ISBN: 1461216249

The fifth and final volume to establish the results claimed by the great Indian mathematician Srinivasa Ramanujan in his "Notebooks" first published in 1957. Although each of the five volumes contains many deep results, the average depth in this volume is possibly greater than in the first four. There are several results on continued fractions - a subject that Ramanujan loved very much. It is the authors wish that this and previous volumes will serve as springboards for further investigations by mathematicians intrigued by Ramanujans remarkable ideas.

The Theory of Jacobi Forms

The Theory of Jacobi Forms
Author: Martin Eichler
Publisher: Springer Science & Business Media
Total Pages: 156
Release: 2013-12-14
Genre: Mathematics
ISBN: 1468491628

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.

Resonance of Ramanujan's Mathematics

Resonance of Ramanujan's Mathematics
Author: Ratan Prakash Agarwal
Publisher: New Age International Limited Publishers
Total Pages: 242
Release: 1996
Genre: Mathematics
ISBN:

This Volume Is In Continuation Of Vol. I And Contains A Critical And Detailed Appraisal Of Ramanujans Work On Theta Functions And Partial Theta Functions, Mock Theta Functions Of Orders Three, Five, Seven And Six (Formally, Designated As Such By Andrews And Hickerson), Lambert Series And Their Relationship With Elliptic Functions, Mock Theta Functions And Allied Functions.A Characteristic Feature Of The Book Is A Detailed Discussion Of The Still Unsettled Problem Of Defining The Order Of A Mock Theta Function And A Discussion Of The Recently Defined Partial Mock Theta Functions And Their Import On Giving New Information On The Structure And Interrelationships Between Mock Theta Functions Of Certain Classes.

Harmonic Maass Forms and Mock Modular Forms: Theory and Applications

Harmonic Maass Forms and Mock Modular Forms: Theory and Applications
Author: Kathrin Bringmann
Publisher: American Mathematical Soc.
Total Pages: 409
Release: 2017-12-15
Genre: Mathematics
ISBN: 1470419440

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.