Elliptic and Modular Functions from Gauss to Dedekind to Hecke

Voorkant
Cambridge University Press, 18 apr. 2017 - 475 pagina's
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This thorough work presents the fundamental results of modular function theory as developed during the nineteenth and early-twentieth centuries. It features beautiful formulas and derives them using skillful and ingenious manipulations, especially classical methods often overlooked today. Starting with the work of Gauss, Abel, and Jacobi, the book then discusses the attempt by Dedekind to construct a theory of modular functions independent of elliptic functions. The latter part of the book explains how Hurwitz completed this task and includes one of Hurwitz's landmark papers, translated by the author, and delves into the work of Ramanujan, Mordell, and Hecke. For graduate students and experts in modular forms, this book demonstrates the relevance of these original sources and thereby provides the reader with new insights into contemporary work in this area.
 

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Inhoudsopgave

The Basic Modular Forms of the Nineteenth Century
1
Hermites Transformation of Theta Functions
5
Gausss Contributions to Modular Forms
13
Abel and Jacobi on Elliptic Functions
42
5
132
6
149
7
188
8
212
The Theory of Modular Forms as Reworked by Hurwitz
334
Ramanujans Euler Products and Modular Forms
344
Dirichlet Series and Modular Forms
371
Sums of Squares
384
The Hecke Operators
426
Translation of Hurwitzs Paper of 1904
445
Bibliography
463
Index
471

The η Function and Dedekind Sums
251
Modular Forms and Invariant Theory
276
The Modular and Multiplier Equations
295

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Over de auteur (2017)

Ranjan Roy is the Huffer Professor of Mathematics and Astronomy at Beloit College, Wisconsin, and has published papers in differential equations, fluid mechanics, complex analysis, and the development of mathematics. He received the Allendoerfer Prize, the Wisconsin MAA teaching award, and the MAA Haimo Award for Distinguished Mathematics Teaching, and was twice named Teacher of the Year at Beloit College. He is a co-author of three chapters in the NIST Handbook of Mathematical Functions, of Special Functions (with Andrews and Askey, Cambridge, 2010), and the author of Sources in the Development of Mathematics (Cambridge, 2011).

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