复分析-第2版 本书特色

　　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.

复分析-第2版 目录

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