Department of Mathematics

Introduction

The topics below will give you a general idea of what is required for the PhD exam in Complex Analysis. The exam may vary a bit year to year depending on who has taught MAA 6406/7 and the text that was used. The most common text is Conway’s Functions of One Complex Variable (Graduate Texts in Mathematics – Vol 11), Second Edition. The topics below are organized by chapters in Conway’s book. Other books may organize things a bit differently, but most of the basic topics will be the same.

The GMA archive contains many old complex exams. These give a good variety of sample problems.

Topics

1. The complex number system
2. Metric spaces and the topology of C. This material forms a background and will not be specifically tested.
3. Elementary Properties and Examples of Analytic Functions
   1. Power series, convergence and basic properties
   2. Analytic functions, definitions and elementary properties.
   3. Sine, Cosine, Exp, and Log in the complex plane.
   4. Cauchy-Riemann equations
   5. Mobius transformations
4. Complex Integration
   1. Line integrals
   2. Power series expansions of analytic functions
   3. Cauchy’s Estimate
   4. Multiplicity of Zeros
   5. Analytic function determined by value on set with accumulation
   6. Liouville’s Theorem
   7. Maximum Modulus Theorem (version one)
   8. Index of a closed curve
   9. Cauchy’s Theorem and Integral Formula (various versions)
   10. The Argument Principle (version 1)
   11. Open Mapping Theorem
5. Singularities
   1. Types of singularities
   2. Order of poles and local form of function
   3. Laurent series
   4. Casorati-Weirstrass Theorem
   5. Residues and the Residue Theorem
   6. Contour integral calculations
   7. The Argument Principle (full version)
   8. Rouche’s Theorem
6. The Maximum Modulus Theorem
   1. The Maximum Modulus Theorem (various versions)
   2. Schwarz’s Lemma
7. Compactness and Convergence in the Space of Analytic Functions
   1. Spaces of continuous functions
   2. Normal families
   3. Equicontinuity
   4. Arzela-Ascoli Theorem
   5. Spaces of analytic functions
   6. Hurwitz’s Theorem
   7. Montel’s Theorem
   8. The Riemann Mapping Theorem
   9. Innite products
   10. The Weierstrass Factorization Theorem
   11. The gamma function
   12. The Riemann zeta function
8. Harmonic Functions
   1. Mean value property
   2. Maximum principle (various versions)
   3. Harmonic functions on a disk
   4. The Poisson kernel
   5. Harnack’s Theorem
   6. Subharmonic and superharmonic functions
   7. Perron families

Additional topics: These might be included depending on who taught your course and/or is making up the exam. Be sure to check.

1. Goursat’s Theorem
2. Convex functions and Hadamard’s Three Circles Theorem
3. The Phragmen-Lindelof Theorem
4. Spaces of meromorphic functions
5. Runge’s Theorem
6. Mittag-Leer’s Theorem
7. Schwarz Reflection Principle
8. Analytic Continuation along a path
9. The Dirichlet Problem
10. Green’s Function
11. Entire functions
12. Little and Great Picard Theorems.

(posted March 26, 2014)