This book is a translation of the forthcoming fourth edition of our German book \"Funktionentheorie P' (Springer 2005). The translation and the LATEX files have been produced by Dan Fulea. He also made a lot of suggestions for improvement which influenced the English version of the book. It is a pleasure for us to express to him our thanks. We also want to thank our colleagues Diarmuid Crowley, Winfried Kohnen and Jorg Sixt for useful suggestions concerning the translation.
Differential Calculus in the Complex Plane C
Ⅰ.1 Complex Numbers
Ⅰ.2 Convergent Sequences and Series
Ⅰ.3 Continuity
Ⅰ.4 Complex Derivatives
Ⅰ.5 The CAUCHY-RIEMANN Differential Equations
Ⅱ Integral Calculus in the Complex Plane C
Ⅱ.1 Complex Line Integrals
Ⅱ.2 The CAUCHY Integral Theorem
Ⅱ.3 The CAUCHY Integral Formulas
Ⅲ Sequences and Series of Analytic Functions, the Residue Theorem
Ⅲ.1 Uniform Approximation
Ⅲ.2 Power Series
Ⅲ.3 Mapping Properties of Analytic Functions
Ⅲ.4 Singularities of Analytic Functions
Ⅲ.5 LAURENT Decomposition A Appendix to III.4 and III.5
Ⅲ.6 The Residue Theorem
Ⅲ.7 Applications of the R,esidue Theorem
Ⅳ Construction of Analytic Functions
Ⅳ.1 The Gamma Function
Ⅳ.2 The WEIERs'rRASS Product Formula
Ⅳ.3 The MITrrAc_LEFFLER Partial FYaction Decomposition
Ⅳ.4 The RIEMANN Mapping Theorem
A Appendix : The Homotopical Version of the CAUCHY Integral Theorem
B Appendix : A Homological Version of the CAUCHY
Integral Theorem
C Appendix : Characterizations of Elementary Domains
Ⅴ Elliptic Functions
Ⅴ.1 LIOUViLLE'S Theorems
A Appendix to the Definition of the Period Lattice
Ⅴ.2 The WEIERSTRASS -function
Ⅴ.3 The Field of Elliptic Functions
A Appendix to Sect. V.3 : The Torus as an Algebraic Curve
Ⅴ.4 The Addition Theorem
Ⅴ.5 Elliptic Integrals
Ⅴ.6 ABEL'S Theorem
Ⅴ.7 The Elliptic Modular Group
Ⅴ.8 The Modular Function j
Ⅵ Elliptic Modular Forms
Ⅵ.1 The Modular Group and Its Fundamental Region
Ⅵ.2 The k/12-formula and the Injectivity of the j-function
Ⅵ.3 The Algebra of Modular Forms
Ⅵ.4 Modular Forms and Theta Series
Ⅵ.5 Modular Forms for Congruence Groups
A Appendix to V1.5 : The Theta Group
Ⅵ.6 A Ring of Theta Fhnctions
Ⅶ Analytic Number Theory
Ⅶ.1 Sums of Four and Eight Squares
Ⅶ.2 DIRiCHLErr Series
Ⅶ.3 DIRICHLET Series with Functional Equations
Ⅶ.4 The RIEMANN <-function and Prime Numbers
Ⅶ.5 The Analytic Continuation of the <-function
Ⅶ.6 A TAUBERian Theorem
Ⅷ Solutions to the Exercises
Ⅷ.1 Solutions to the Exercises of Chapter Ⅰ
Ⅷ.2 Solutions to the Exercises of Chapter Ⅱ
Ⅷ.3 Solutions to the Exercises of Chapter Ⅲ
Ⅷ.4 Solutions to the Exercises of Chapter Ⅳ
Ⅷ.5 Solutions to the Exercises of Chapter Ⅴ
Ⅷ.6 Solutions to the Exercises of Chapter Ⅵ
Ⅷ.7 Solutions to the Exercises of Chapter Ⅶ
R/eferences
Symbolic Notations
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