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Differential Calculus in the Complex Plane CⅠ.1 Complex NumbersⅠ.2 Convergent Sequences and SeriesⅠ.3 ContinuityⅠ.4 Complex DerivativesⅠ.5 The CAUCHY-RIEMANN Differential EquationsⅡ Integral Calculus in the Complex Plane CⅡ.1 Complex Line IntegralsⅡ.2 The CAUCHY Integral TheoremⅡ.3 The CAUCHY Integral FormulasⅢ　Sequences and Series of Analytic Functions， the Residue　TheoremⅢ.1 Uniform ApproximationⅢ.2 Power SeriesⅢ.3 Mapping Properties of Analytic FunctionsⅢ.4 Singularities of Analytic FunctionsⅢ.5 LAURENT Decomposition　A Appendix to III.4 and III.5 Ⅲ.6 The Residue TheoremⅢ.7 Applications of the R，esidue TheoremⅣ Construction of Analytic FunctionsⅣ.1 The Gamma FunctionⅣ.2 The WEIERs'rRASS Product FormulaⅣ.3 The MITrrAc_LEFFLER Partial FYaction DecompositionⅣ.4 The RIEMANN Mapping TheoremA Appendix ： The Homotopical Version of the CAUCHY　Integral TheoremB Appendix ： A Homological Version of the CAUCHYIntegral TheoremC Appendix ： Characterizations of Elementary DomainsⅤ Elliptic FunctionsⅤ.1 LIOUViLLE'S TheoremsA Appendix to the Definition of the Period LatticeⅤ.2 The WEIERSTRASS -functionⅤ.3 The Field of Elliptic FunctionsA Appendix to Sect. V.3 ： The Torus as an Algebraic CurveⅤ.4 The Addition TheoremⅤ.5 Elliptic IntegralsⅤ.6 ABEL'S TheoremⅤ.7 The Elliptic Modular GroupⅤ.8 The Modular Function jⅥ Elliptic Modular FormsⅥ.1 The Modular Group and Its Fundamental RegionⅥ.2 The k/12-formula and the Injectivity　of the j-functionⅥ.3 The Algebra of Modular FormsⅥ.4 Modular Forms and Theta SeriesⅥ.5 Modular Forms for Congruence GroupsA Appendix to V1.5 ： The Theta GroupⅥ.6 A Ring of Theta FhnctionsⅦ　Analytic Number TheoryⅦ.1 Sums of Four and Eight SquaresⅦ.2 DIRiCHLErr SeriesⅦ.3 DIRICHLET Series with Functional EquationsⅦ.4 The RIEMANN

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.