Complex Analysis

by Freitag, Eberhard; Busam, Rolf

• ISBN13: 9783540939825
• ISBN10: 3540939822
• Edition: 2nd
• Format: Paperback
• Copyright: 2009-05-29
• Publisher: Springer Nature

Summary

The idea of this book is to give an extensive description of the classical complex analysis, here ''classical'' means roughly that sheaf theoretical and cohomological methods are omitted.The first four chapters cover the essential core of complex analysis presenting their fundamental results. After this standard material, the authors step forward to elliptic functions and to elliptic modular functions including a taste of all most beautiful results of this field. The book is rounded by applications to analytic number theory including distinguished pearls of this fascinating subject as for instance the Prime Number Theorem. Great importance is attached to completeness, all needed notions are developed, only minimal prerequisites (elementary facts of calculus and algebra) are required.More than 400 exercises including hints for solutions and many figures make this an attractive, indispensable book for students who would like to have a sound introduction to classical complex analysis.For the second edition the authors have revised the text carefully.

Author Biography

Eberhard Freitag is full professor at the University of Heidelberg. His field of research is the theory of modular functions. He wrote several monographs about this subject. Rolf Busam received his PhD in Mathematics at the University of Heidelberg. His current interests include Complex Analysis with applications to Analytic Number Theory, Automorphic Forms and Computer Algebra.

Table of Contents

Differential Calculus in the Complex Plane C	p. 9
Complex Numbers	p. 9
Convergent Sequences and Series	p. 24
Continuity	p. 36
Complex Derivatives	p. 42
The Cauchy-Riemann Differential Equations	p. 47
Integral Calculus in the Complex Plane C	p. 69
Complex Line Integrals	p. 70
The Cauchy Integral Theorem	p. 77
The Cauchy Integral Formulas	p. 92
Sequences and Series of Analytic Functions, the Residue Theorem	p. 103
Uniform Approximation	p. 104
Power Series	p. 109
Mapping Properties of Analytic Functions	p. 124
Singularities of Analytic Functions	p. 133
Laurent Decomposition	p. 142
Appendix to III.4 and III.5	p. 155
The Residue Theorem	p. 162
Applications of the Residue Theorem	p. 170
Construction of Analytic Functions	p. 191
The Gamma Function	p. 192
The Weierstrass Product Formula	p. 210
The Mittag-Leffler Partial Fraction Decomposition	p. 218
The Riemann Mapping Theorem	p. 223
Appendix : The Homotopical Version of the Cauchy Integral Theorem	p. 233
Appendix : A Homological Version of the Cauchy Integral Theorem	p. 239
Appendix : Characterizations of Elementary Domains	p. 244
Elliptic Functions	p. 251
Liouville's Theorems	p. 252
Appendix to the Definition of the Period Lattice	p. 259
The Weierstrass $$-function	p. 261
The Field of Elliptic Functions	p. 267
Appendix to Sect. V.3 : The Torus as an Algebraic Curve	p. 271
The Addition Theorem	p. 278
Elliptic Integrals	p. 284
Abel's Theorem	p. 291
The Elliptic Modular Group	p. 301
The Modular Function j	p. 309
Elliptic Modular Forms	p. 317
The Modular Group and Its Fundamental Region	p. 318
The k/12-formula and the Injectivity of the j-function	p. 326
The Algebra of Modular Forms	p. 334
Modular Forms and Theta Series	p. 338
Modular Forms for Congruence Groups	p. 352
Appendix to VI.5 : The Theta Group	p. 363
A Ring of Theta Functions	p. 370
Analytic Number Theory	p. 381
Sums of Four and Eight Squares	p. 382
Dirichlet Series	p. 399
Dirichlet Series with Functional Equations	p. 408
The Riemann ¿-function and Prime Numbers	p. 421
The Analytic Continuation of the ¿-function	p. 429
A Tauberian Theorem	p. 436
Solutions to the Exercises	p. 449
Solutions to the Exercises of Chapter I	p. 449
Solutions to the Exercises of Chapter II	p. 459
Solutions to the Exercises of Chapter III	p. 464
Solutions to the Exercises of Chapter IV	p. 475
Solutions to the Exercises of Chapter V	p. 482
Solutions to the Exercises of Chapter VI	p. 490
Solutions to the Exercises of Chapter VII	p. 498
References	p. 509
Symbolic Notations	p. 519
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