Table of Contents

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 ? of ?.- 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 1-Forms Along Continuous Paths.- 1.8. Index of a Loop.- 1.9. Homology.- 1.10. Residues.- 1.11. Holomorphic Functions.- 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.- 3 The $$bar partial$$-Equation.- 3,1. Runge’s Theorem.- 3.2. Mittag—Leffler’s Theorem.- 3.3. The Weierstrass Theorem.- 3.4. An Interpolation Theorem.- 3.5. Closed Ideals in ? (?).- 3.6. The Operator $$frac{partial }{{partial bar z}}$$ Acting on Distributions.- 3.7. Mergelyan’s Theorem.- 3.8. Short Survey of the Theory of Distributions. Their Relation to the Theory of Residues.- 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 ?.- 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.- 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 ? 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.