preface
chapter 1 topology of the complex plane and holomorphic
functions
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 stokes
formula
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 to
thetheory 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
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