Complex analysis / Eberhard Freitag, Rolf Busam.

Format:

Book

Author/Creator:

Freitag, E. (Eberhard)

Contributor:

Busam, Rolf.

Series:

Universitext

Standardized Title:

Funktionentheorie. English

Language:

English

German

Subjects (All):

Functions of complex variables.

Functions of complex variables--Problems, exercises, etc.

Genre:

Problems and exercises.

Physical Description:

x, 532 pages : illustrations ; 24 cm.

Edition:

Second edition, [Second English edition].

Place of Publication:

Berlin : Springer, [2009]

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 perequisites (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.

Contents:

I Differential Calculus in the Complex Plane C 9

I.1 Complex Numbers 9

I.2 Convergent Sequences and Series 24

I.3 Continuity 36

I.4 Complex Derivatives 42

I.5 The Cauchy-Riemann Differential Equations 47

II Integral Calculus in the Complex Plane C 69

II.1 Complex Line Integrals 70

II.2 The Cauchy Integral Theorem 77

II.3 The Cauchy Integral Formulas 92

III Sequences and Series of Analytic Functions, the Residue Theorem 103

III.1 Uniform Approximation 104

III.2 Power Series 109

III.3 Mapping Properties of Analytic Functions 124

III.4 Singularities of Analytic Functions 133

III.5 Laurent Decomposition 142

A Appendix to III.4 and III.5 155

III.6 The Residue Theorem 162

III.7 Applications of the Residue Theorem 170

IV Construction of Analytic Functions 191

IV.1 The Gamma Function 192

IV.2 The Weierstrass Product Formula 210

IV.3 The Mittag-Leffler Partial Fraction Decomposition 218

IV.4 The Riemann Mapping Theorem 223

A Appendix : The Homotopical Version of the Cauchy Integral Theorem 233

B Appendix : A Homological Version of the Cauchy Integral Theorem 239

C Appendix : Characterizations of Elementary Domains 244

V Elliptic Functions 251

V.1 Liouville's Theorems 252

A Appendix to the Definition of the Period Lattice 259

V.2 The Weierstrass $$-function 261

V.3 The Field of Elliptic Functions 267

A Appendix to Sect. V.3 : The Torus as an Algebraic Curve 271

V.4 The Addition Theorem 278

V.5 Elliptic Integrals 284

V.6 Abel's Theorem 291

V.7 The Elliptic Modular Group 301

V.8 The Modular Function j 309

VI Elliptic Modular Forms 317

VI.1 The Modular Group and Its Fundamental Region 318

VI.2 The k/12-formula and the Injectivity of the j-function 326

VI.3 The Algebra of Modular Forms 334

VI.4 Modular Forms and Theta Series 338

VI.5 Modular Forms for Congruence Groups 352

A Appendix to VI.5 : The Theta Group 363

VI.6 A Ring of Theta Functions 370

VII Analytic Number Theory 381

VII.1 Sums of Four and Eight Squares 382

VII.2 Dirichlet Series 399

VII.3 Dirichlet Series with Functional Equations 408

VII.4 The Riemann ζ-function and Prime Numbers 421

VII.5 The Analytic Continuation of the ζ-function 429

VII.6 A Tauberian Theorem 436

VIII Solutions to the Exercises 449

VIII.1 Solutions to the Exercises of Chapter I 449

VIII.2 Solutions to the Exercises of Chapter II 459

VIII.3 Solutions to the Exercises of Chapter III 464

VIII.4 Solutions to the Exercises of Chapter IV 475

VIII.5 Solutions to the Exercises of Chapter V 482

VIII.6 Solutions to the Exercises of Chapter VI 490

VIII.7 Solutions to the Exercises of Chapter VII 498.

Notes:

Previous ed.: 2005.

Includes bibliographical references (pages [509]-517) and index.

ISBN:

9783540939825

3540939822

3540939830

9783540939832

OCLC:

316434231