复变导论

prefacechapter 1 topology of the complex plane and holomorphicfunctions1.1. some linear algebra and differential calculus1.2. differential forms on an open subset fl of c1.3. partitions of unity1.4. regular boundaries1.5. integration of differential forms of degree 2. the stokesformula1.6. homotopy. fundamental group1.7. integration of closed i-forms along continuous paths1.8. index of a loop1.9. homology1.10. residues1.11. holomorphic functionschapter 2 analytic properties of holomorphic functions2.1. integral representation formulas2.2. the frechet space2.3. holomorphic maps2.4. isolated singularities and residues2.5. residues and the computation of definite integrals2.6. other applications of the residue theorem2.7. the area theorem2.8. conformal mappingschapter 3 the -equation3.1. runge'stheorem3.2. mittag-leffier's theorem3.3. the weierstrass theorem3.4. an interpolation theorem3.5. closed ideals in (ω)3.6. the operator σ acting on distributions3.7. mergelyan's theorem3.8. short survey of the theory of distributions. their relation tothetheory of residueschapter 4 harmonic and subharmonic functions4.1. introduction4.2. a remark on the theory of integration4.3. harmonic functions4.4. subharmonic functions4.5. order and type of subharmonic functions in c4.6. integral representations4.7. green functions and harmonic measure4.8. smoothness up to the boundary of biholomorphic mappings4.9. introduction to potential theorychapter 5 analytic continuation and singularities5.1. introduction5.2. elementary study of singularities and dirichlet series5.3. a brief study of the functions f and5.4. covering spaces5.5. riemann surfaces5.6. the sheaf of germs of holomorphic functions5.7. cocycles5.8. group actions and covering spaces5.9. galois coverings5.10 the exact sequence of a galois covering5.11. universal covering space5.12. algebraic functions, i5.13. algebraic functions, ii5.14. the periods of a differential form5.15. linear differential equations5.16. the index of differential operatorsreferencesnotation and selected terminologyindex