复变导论

　　贝莱恩斯坦编著的《复变导论（英文影印版）》给出了一个全纯函数性质的概述。内容全面，囊括了微分形式、同伦理论、同调理论和全纯函数的解析性质、非同质的Cauchy-Riemann方程的可解性和子调和函数理论，引入层理论、覆盖空间和黎曼曲面。为了帮助读者更好地理解书中的材料，增加了大量不同难度的习题。

preface
chapter 1 topology of the complex plane and holomorphicfunctions
1.1. some linear algebra and differential calculus
1.2. differential forms on an open subset fl of c
1.3. partitions of unity
1.4. regular boundaries
1.5. integration of differential forms of degree 2. the stokesformula
1.6. homotopy. fundamental group
1.7. integration of closed i-forms along continuous paths
1.8. index of a loop
1.9. homology
1.10. residues
1.11. holomorphic functions
chapter 2 analytic properties of holomorphic functions
2.1. integral representation formulas
2.2. the frechet space
2.3. holomorphic maps
2.4. isolated singularities and residues
2.5. residues and the computation of definite integrals
2.6. other applications of the residue theorem
2.7. the area theorem
2.8. conformal mappings
chapter 3 the -equation
3.1. runge’stheorem
3.2. mittag-leffier’s theorem
3.3. the weierstrass theorem
3.4. an interpolation theorem
3.5. closed ideals in (ω)
3.6. the operator σ acting on distributions
3.7. mergelyan’s theorem
3.8. short survey of the theory of distributions. their relation tothetheory of residues
chapter 4 harmonic and subharmonic functions
4.1. introduction
4.2. a remark on the theory of integration
4.3. harmonic functions
4.4. subharmonic functions
4.5. order and type of subharmonic functions in c
4.6. integral representations
4.7. green functions and harmonic measure
4.8. smoothness up to the boundary of biholomorphic mappings
4.9. introduction to potential theory
chapter 5 analytic continuation and singularities
5.1. introduction
5.2. elementary study of singularities and dirichlet series
5.3. a brief study of the functions f and
5.4. covering spaces
5.5. riemann surfaces
5.6. the sheaf of germs of holomorphic functions
5.7. cocycles
5.8. group actions and covering spaces
5.9. galois coverings
5.10 the exact sequence of a galois covering
5.11. universal covering space
5.12. algebraic functions, i
5.13. algebraic functions, ii
5.14. the periods of a differential form
5.15. linear differential equations
5.16. the index of differential operators
references
notation and selected terminology
index