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El. knyga: Elliptic and Modular Functions from Gauss to Dedekind to Hecke

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(Beloit College, Wisconsin)
  • Formatas: PDF+DRM
  • Išleidimo metai: 18-Apr-2017
  • Leidėjas: Cambridge University Press
  • Kalba: eng
  • ISBN-13: 9781108133784
Kitos knygos pagal šią temą:
Elliptic and Modular Functions from Gauss to Dedekind to HeckeDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Išleidimo metai: 18-Apr-2017
  • Leidėjas: Cambridge University Press
  • Kalba: eng
  • ISBN-13: 9781108133784
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Many mathematicians today believe mathematical works from past centuries are inaccessible, not useful, or both. This book demonstrates that past works are not only readable, but brimming with ideas and insights. It is aimed at graduate students or researchers working in modular functions, number theory, complex analysis, or special functions.

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.

Recenzijos

'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 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

Daugiau informacijos

A thorough guide to elliptic functions and modular forms that demonstrates the relevance and usefulness of historical sources.
Preface ix
1 The Basic Modular Forms of the Nineteenth Century
1(12)
1.1 The Modular Group
1(4)
1.2 Modular Forms
5(6)
1.3 Exercises
11(2)
2 Gauss's Contributions to Modular Forms
13(29)
2.1 Early Work on Elliptic Integrals
13(4)
2.2 Landen and Legendre's Quadratic Transformation
17(1)
2.3 Lagrange's Arithmetic-Geometric Mean
18(2)
2.4 Gauss on the Arithmetic-Geometric Mean
20(7)
2.5 Gauss on Elliptic Functions
27(5)
2.6 Gauss: Theta Functions and Modular Forms
32(4)
2.7 Exercises
36(6)
3 Abel and Jacobi on Elliptic Functions
42(52)
3.1 Preliminary Remarks
42(12)
3.2 Jacobi on Transformations of Orders 3 and 5
54(6)
3.3 The Jacobi Elliptic Functions
60(3)
3.4 Transformations of Order n and Infinite Products
63(3)
3.5 Jacobi's Transformation Formulas
66(4)
3.6 Equivalent Forms of the Transformation Formulas
70(1)
3.7 The First and Second Transformations
71(1)
3.8 Complementary Transformations
72(2)
3.9 Jacobi's First Supplementary Transformation
74(1)
3.10 Jacobi's Infinite Products for Elliptic Functions
75(5)
3.11 Jacobi's Theory of Theta Functions
80(6)
3.12 Jacobi's Triple Product Identity
86(3)
3.13 Modular Equations and Transformation Theory
89(1)
3.14 Exercises
90(4)
4 Eisenstein and Hurwitz
94(38)
4.1 Preliminary Remarks
94(7)
4.2 Eisenstein's Theory of Trigonometric Functions
101(4)
4.3 Eisenstein's Derivation of the Addition Formula
105(1)
4.4 Eisenstein's Theory of Elliptic Functions
106(3)
4.5 Differential Equations for Elliptic Functions
109(4)
4.6 The Addition Theorem for the Elliptic Function
113(2)
4.7 Eisenstein's Double Product
115(1)
4.8 Elliptic Functions in Terms of the ø Function
116(1)
4.9 Connection of ø with Theta Functions
117(6)
4.10 Hurwitz's Fourier Series for Modular Forms
123(3)
4.11 Hurwitz's Proof That Δ(ω) Is a Modular Form
126(2)
4.12 Hurwitz's Proof of Eisenstein's Result
128(1)
4.13 Kronecker's Proof of Eisenstein's Result
129(1)
4.14 Exercises
130(2)
5 Hermite's Transformation of Theta Functions
132(17)
5.1 Preliminary Remarks
132(6)
5.2 Hermite's Proof of the Transformation Formula
138(3)
5.3 Smith on Jacobi's Formula for the Product of Four Theta Functions
141(6)
5.4 Exercises
147(2)
6 Complex Variables and Elliptic Functions
149(39)
6.1 Historical Remarks on the Roots of Unity
149(12)
6.2 Simpson and the Ladies Diary
161(3)
6.3 Development of Complex Variables Theory
164(8)
6.4 Hermite: Complex Analysis in Elliptic Functions
172(4)
6.5 Riemann: Meaning of the Elliptic Integral
176(6)
6.6 Weierstrass's Rigorization
182(2)
6.7 The Phragmen-Lindelof Theorem
184(4)
7 Hypergeometric Functions
188(24)
7.1 Preliminary Remarks
188(1)
7.2 Stirling
189(2)
7.3 Euler and the Hypergeometric Equation
191(1)
7.4 Pfaff's Transformation
192(1)
7.5 Gauss and Quadratic Transformations
193(3)
7.6 Kummer on the Hypergeometric Equation
196(2)
7.7 Riemann and the Schwarzian Derivative
198(3)
7.8 Riemann and the Triangle Functions
201(1)
7.9 The Ratio of the Periods K'/K as a Conformal Map
202(5)
7.10 Schwarz: Hypergeometric Equation with Algebraic Solutions
207(3)
7.11 Exercises
210(2)
8 Dedekind's Paper on Modular Functions
212(39)
8.1 Preliminary Remarks
212(4)
8.2 Dedekind's Approach
216(3)
8.3 The Fundamental Domain for SL2 (Z)
219(3)
8.4 Tesselation of the Upper Half-plane
222(1)
8.5 Dedekind's Valency Function
222(1)
8.6 Branch Points
223(2)
8.7 Differential Equations
225(3)
8.8 Dedekind's η Function
228(6)
8.9 The Uniqueness of k2
234(1)
8.10 The Connection of η with Theta Functions
234(1)
8.11 Hurwitz's Infinite Product for η(ω)
235(1)
8.12 Algebraic Relations among Modular Forms
236(2)
8.13 The Modular Equation
238(5)
8.14 Singular Moduli and Quadratic Forms
243(6)
8.15 Exercises
249(2)
9 The η Function and Dedekind Sums
251(25)
9.1 Preliminary Remarks
251(7)
9.2 Riemann's Notes
258(6)
9.3 Dedekind Sums in Terms of a Periodic Function
264(5)
9.4 Rademacher
269(5)
9.5 Exercises
274(2)
10 Modular Forms and Invariant Theory
276(19)
10.1 Preliminary Remarks
276(3)
10.2 The Early Theory of Invariants
279(6)
10.3 Cayley's Proof of a Result of Abel
285(2)
10.4 Reduction of an Elliptic Integral to Riemann's Normal Form
287(2)
10.5 The Weierstrass Normal Form
289(2)
10.6 Proof of the Infinite Product for Δ
291(2)
10.7 The Multiplier in Terms of 12√Δ
293(2)
11 The Modular and Multiplier Equations
295(39)
11.1 Preliminary Remarks
295(8)
11.2 Jacobi's Multiplier Equation
303(1)
11.3 Sohnke's Paper on Modular Equations
304(10)
11.4 Brioschi on Jacobi's Multiplier Equation
314(3)
11.5 Joubert on the Multiplier Equation
317(3)
11.6 Kiepert and Klein on the Multiplier Equation
320(6)
11.7 Hurwitz: Roots of the Multiplier Equation
326(6)
11.8 Exercises
332(2)
12 The Theory of Modular Forms as Reworked by Hurwitz
334(10)
12.1 Preliminary Remarks
334(1)
12.2 The Fundamental Domain
335(1)
12.3 An Infinite Product as a Modular Form
336(3)
12.4 The J-Function
339(3)
12.5 An Application to the Theory of Elliptic Functions
342(2)
13 Ramanujan's Euler Products and Modular Forms
344(27)
13.1 Preliminary Remarks
344(4)
13.2 Ramanujan's τ Function
348(2)
13.3 Ramanujan: Product Formula for Δ
350(3)
13.4 Proof of Identity (13.2)
353(3)
13.5 The Arithmetic Function τ(n)
356(6)
13.6 Mordell on Euler Products
362(5)
13.7 Exercises
367(4)
14 Dirichlet Series and Modular Forms
371(13)
14.1 Preliminary Remarks
371(2)
14.2 Functional Equations for Dirichlet Series
373(7)
14.3 Theta Series in Two Variables
380(2)
14.4 Exercises
382(2)
15 Sums of Squares
384(42)
15.1 Preliminary Remarks
384(9)
15.2 Jacobi's Elliptic Functions Approach
393(1)
15.3 Glaisher
394(3)
15.4 Ramanjuan's Arithmetical Functions
397(3)
15.5 Mordell: Spaces of Modular Forms
400(5)
15.6 Hardy's Singular Series
405(5)
15.7 Hecke's Solution to the Sums of Squares Problem
410(14)
15.8 Exercises
424(2)
16 The Hecke Operators
426(19)
16.1 Preliminary Remarks
426(2)
16.2 The Hecke Operators T(n)
428(6)
16.3 The Operators T(n) in Terms of Matrices λ(n)
434(4)
16.4 Euler Products
438(1)
16.5 Eigenfunctions of the Hecke Operators
439(3)
16.6 The Petersson Inner Product
442(2)
16.7 Exercises
444(1)
Appendix: Translation of Hurwitz's Paper of 1904
445(18)
§1 Equivalent Quantities
445(3)
§2 The Modular Forms Gn(ω1, ω2)
448(4)
§3 The Representation of the Function Gn by Power Series
452(2)
§4 The Modular Form Δ(ω1, ω2)
454(1)
§5 The Modular Function J(ω)
455(5)
§6 Applications to the Theory of Elliptic Functions
460(3)
Bibliography 463(8)
Index 471
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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