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| 內容簡介： |
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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.
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| 目錄：
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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 k12-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
References
Symbolic Notations
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