A Second Course in Complex Analysis

1. Preliminaries

1.1 Holomorphic Functions

1.2 Meromorphic Functions

1.3 Algebraic Functions

1.4 The Mittag-Leffler Problem in an Open Planar Domain

1.5 The Compact Support Case

1.6 The General Case

1.7 Paths and Curves

1.8 Continuous Choice of Argument

1.9 Homotopic Curves

1.10 Index of a Loop

1.11 The Fundamental Group

1.12 Connectivity

1.13 The Brouwer Fixed Point Theorem

1.14 Gamma Function

2. Extremal Length

2.1 Some Definitions

2.2 The Conformal Invariance of Extremal Length

2.3 Further Properties of Extremal Lengths

2.4 Some Examples

3. Harmonic Measure

3.1 The Idea of Harmonic Measure

3.2 A Discussion of Interpolation of Linear Operators

3.3 The F. and M. Riesz Theorem

4. Riemann Surfaces

4.1 The Riemann Surface of a Function

4.2 Constructions of Riemann Surfaces

4.3 Analytic Continuation of Complex Functions

4.4 Riemann Surfaces of Functions

4.4.5 Global Topological Properties

4.5 Puiseux Series

4.6 Algebraic Curves

4.7 The Connectedness of Algebraic Curves

4.8 Riemann Surfaces Associated to a Polynomial

4.9 Smooth Projective Plane Curves

4.10 Compact Riemann Surfaces

4.11 Projective Algebraic Curves

4.12 Projective Closure and Affine Restriction

4.13 Example 4.4.12 (continuation)

4.14 Smooth and Singular Points on Affine and Projective Curves

4.15 Chow’s Theorem

5. Abstract Riemann Surfaces

5.1 Basic Definitions

5.2 Examples of Riemann Surfaces

5.3 Genus of a Compact Riemann Surfaces

5.4 Triangulations of Riemann Surfaces

5.5 A Short Proof that Compact 2-Manifolds Can Be Triangulated

5.6 Holomorphic and Meromorphic Functions

5.7 Elliptic Functions

5.8 Rational Functions on Riemann Surfaces

5.9 The Argument Principle on Riemann Surfaces

5.10 Construction Conceived by Riemann

5.11 The Riemann–Hurwitz Formula — Applications

5.12 Any Two Meromorphic Functions are Algebraically Related

5.13 Holomorphic and Meromorphic Differential Form

5.14 Harmonic Differential Forms

5.15 The Dimension of the Space of Holomorphic Differential Forms Ω1(X)

5.16 Every Riemann Surface Admits a Non-constant Meromorphic Function

6. The Riemann–Roch Theorem

6.1 Introduction

6.2 Baby Proof of the Riemann–Roch Theorem

6.3 The Residue Theorem on Compact Riemann Surfaces

6.4 Dependence Among the Linear Constraints

6.5 Next Steps Towards the Riemann–Roch Theorem

6.6 Higher Order Poles

6.7 Divisors

6.8 The Riemann–Roch Problem in Terms of Divisors

6.9 The Canonical Divisor

6.10 The Riemann–Roch Formula

6.11 The Geometric Riemann–Roch Theorem

6.12 Embedding into Projective Space

6.13 Compact Riemann Surfaces and Algebraic Curves

6.14 Hyperelliptic Integral

6.15 Abel’s Theorem

6.16 The Jacobi Inversion Problem

7. Covering Surfaces and Classical Plane Geometries

7.1 Normal Families and Automorphisms

7.2 The Basic Examples

7.3 Riemann Surfaces and Covering Spaces

7.4 Covering Spaces and Invariant Metrics, I: Quotients of C

7.5 Covering Spaces and Invariant Metrics, II: Quotients of D

7.6 Compact Quotients of D and Their Automorphisms

7.7 The Automorphism Group of a Riemann Surface of Genus at Least 2

7.8 Automorphisms of Multiply Connected Domains

8. The Uniformization Theorem

8.1 Preliminary Results

8.2 Green’s Functions

8.3 Bipolar Green’s Functions

9. Analytic Capacity

9.1 Calculating Analytic Capacity

9.2 Analytic Capacity and Removability

10. The Bergman Kernel

10.1 Smoothness to the Boundary of KΩ

10.2 Calculating the Bergman Kernel

10.3 The Poincaré-Bergman Metric on the Disc

10.4 Appendix to Section 10.2: The Biholomorphic Inequivalence of the Ball and the Polydisc

11. Appendix

11.1 Covering Spaces

11.2 The Sheaf of Germs of Holomorphic Functions

11.3 Sheaves

11.4 Polynomials

Closing Remarks

Biography

Peter V. Dovbush, Dr. habil., is an Associate Professor at Moldova State University, in the Institute of Mathematics and Computer Science. He received his Ph.D. in Lomonosov Moscow State University in 1983 and Doctor of Sciences in 2003. He has published over 50 scholarly articles.

Steven G. Krantz is a Professor of Mathematics at Washington University in St. Louis.  He has previously taught at UCLA, Princeton University, and Penn State University.  He received his Ph.D. from Princeton University in 1974.  Krantz has directed 20 Ph.D. students and 8 Masters students. He has published over 130 books and over 300 scholarly articles.  He is the holder of the Chauvenet Prize and the Beckenbach Book Award and the Kemper Prize.  He is a Fellow of the American Mathematical Society.