Complex Analysis in one Variable

This book is based on a first-year graduate course I gave three times at the University of Chicago. As it was addressed to graduate students who intended to specialize in mathematics, I tried to put the classical theory of functions of a complex variable in context, presenting proofs and points of view which relate the subject to other branches of mathematics. Complex analysis in one variable is ideally suited to this attempt. Of course, the branches of mathema­ tics one chooses, and the connections one makes, must depend on personal taste and knowledge. My own leaning towards several complex variables will be apparent, especially in the notes at the end of the different chapters. The first three chapters deal largely with classical material which is avai­ lable in the many books on the subject. I have tried to present this material as efficiently as I could, and, even here, to show the relationship with other branches of mathematics. Chapter 4 contains a proof of Picard's theorem; the method of proof I have chosen has far-reaching generalizations in several complex variables and in differential geometry. The next two chapters deal with the Runge approximation theorem and its many applications. The presentation here has been strongly influenced by work on several complex variables.

1 Elementary Theory of Holomorphic Functions. - 1 Some basic properties of ? -differentiable and holomorphic functions. - 2 Integration along curves. - 3 Fundamental properties of holomorphic functions. - 4 The theorems of Weierstrass and Montel. - 5 Meromorphic functions. - 6 The Looman-Menchoff theorem. - Notes on Chapter 1. - References : Chapter 1. - 2 Covering Spaces and the Monodromy Theorem. - 1 Covering spaces and the lifting of curves. - 2 The sheaf of germs of holomorphic functions. - 3 Covering spaces and integration along curves. - 4 The monodromy theorem and the homotopy form of Cauchy s theorem. - 5 Applications of the monodromy theorem. - Notes on Chapter 2. - References : Chapter 2. - 3 The Winding Number and the Residue Theorem. - 1 The winding number. - 2 The residue theorem. - 3 Applications of the residue theorem. - Notes on Chapter 3. - References : Chapter 3. - 4 Picard s Theorem. - Notes on Chapter 4. - References : Chapter 4. - 5 The Inhomogeneous Cauchy-Riemann Equation and Runge s Theorem. - 1 Partitions of unity. - 2 The equation % MathType! MTEF! 2! 1! +-
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\[\frac{{\partial u}}
{{\partial \bar z}} = \phi \]$$. - 3 Runge s theorem. - 4 The homology form of Cauchy s theorem. - Notes on Chapter 5. - References : Chapter 5. - 6 Applications of Runge s Theorem. - 1 The Mittag-Leffler theorem. - 2 The cohomology form of Cauchy s theorem. - 3 The theorem of Weierstrass. - 4 Ideals in ? (?). - Notes on Chapter 6. - References: Chapter 6. - 7 The Riemann Mapping Theorem and Simple Connectedness in the Plane. - 1 Analytic automorphisms of the disc and of the annulus. - 2 The Riemann mapping theorem. - 3 Simply connected plane domains. - Notes on Chapter 7. - References : Chapter 7. - 8 Functions of Several Complex Variables. - Notes on Chapter 8. - References : Chapter 8. - 9 Compact Riemann Surfaces. - 1 Definitions and basic theorems. - 2 Meromorphic functions. - 3 The cohomology group H1( ). - 4 A theorem from functional analysis. - 5 The finiteness theorem. - 6 Meromorphic functions on a compact Riemann surface. - Notes on Chapter 9. - References : Chapter 9. - 10 The Corona Theorem. - 1 The Poisson Integral and the theorem of F and M Riesz. - 2 The corona theorem. - Notes on Chapter 10. - References: Chapter 10. - 11 Subharmonic Functions and the Dirichlet Problem. - 1 Semi-continuous functions. - 2 Harmonic functions and Harnack s principle. - 3 Convex functions. - 4 Subharmonic functions : Definition and basic properties. - 5 Subharmonic functions : Further properties and application to convexity theorems. - 6 Harmonic and subharmonic functions on Riemann surfaces. - 7 The Dirichlet problem. - 8 The Radó-Cartan theorem. - Notes on Chapter 11. - References : Chapter 11.