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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.Read more
- Features detailed analysis of lost or little known methods and techniques used by Gauss, Jacobi, Riemann, Dedekind, Hurwitz, and others
- A translation of Hurwitz's 1904 paper, not easily available in English, is included as an appendix
- Exercises at the end of each chapter allow readers to extend their grasp of the material
Reviews & endorsements
'Finally, it needs to be stressed that Roy does much more than present these mathematical works as museum pieces. He takes pains to tie them in to modern work when reasonable and appropriate, and that of course just adds to the quality of his work. I am very excited to have a copy of this wonderful book in my possession.' Michael Berg, MAA ReviewsSee more reviews
'This book will be a valuable resource for understanding modular functions in their historical context, especially for readers not fluent in the languages of the original papers.' Paul M. Jenkins, Mathematical Reviews
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- Date Published: April 2017
- format: Hardback
- isbn: 9781107159389
- length: 488 pages
- dimensions: 261 x 182 x 31 mm
- weight: 1.02kg
- contains: 13 b/w illus.
- availability: In stock
Table of Contents
1. The basic modular forms
2. Gauss's contributions to modular forms
3. Abel and Jacobi on elliptic functions
4. Eisenstein and Hurwitz
5. Hermite's transformation of theta functions
6. Complex variables and elliptic functions
7. Hypergeometric functions
8. Dedekind's paper on modular functions
9. The n function and Dedekind sums
10. Modular forms and invariant theory
11. The modular and multiplier equations
12. The theory of modular forms as reworked by Hurwitz
13. Ramanujan's Euler products and modular forms
14. Dirichlet series and modular forms
15. Sums of squares
16. The Hecke operators.
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