Complex Variables

Textbooks, even excellent ones, are a reflection of their times. Form and content of books depend on what the students know already, what they are expected to learn, how the subject matter is regarded in relation to other divisions of mathematics, and even how fashionable the subject matter is. It is thus not surprising that we no longer use such masterpieces as Hurwitz and Courant's Funktionentheorie or Jordan's Cours d'Analyse in our courses. The last two decades have seen a significant change in the techniques used in the theory of functions of one complex variable. The important role played by the inhomogeneous Cauchy-Riemann equation in the current research has led to the reunification, at least in their spirit, of complex analysis in one and in several variables. We say reunification since we think that Weierstrass, Poincare, and others (in contrast to many of our students) did not consider them to be entirely separate subjects. Indeed, not only complex analysis in several variables, but also number theory, harmonic analysis, and other branches of mathematics, both pure and applied, have required a reconsidera tion of analytic continuation, ordinary differential equations in the complex domain, asymptotic analysis, iteration of holomorphic functions, and many other subjects from the classic theory of functions of one complex variable. This ongoing reconsideration led us to think that a textbook incorporating some of these new perspectives and techniques had to be written.

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.