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
- The complex number system
- Metric spaces and the topology of C. This material forms a background and will not be specifically tested.
- Elementary Properties and Examples of Analytic Functions
- Power series, convergence and basic properties
- Analytic functions, definitions and elementary properties.
- Sine, Cosine, Exp, and Log in the complex plane.
- Cauchy-Riemann equations
- Mobius transformations
- Complex Integration
- Line integrals
- Power series expansions of analytic functions
- Cauchy’s Estimate
- Multiplicity of Zeros
- Analytic function determined by value on set with accumulation
- Liouville’s Theorem
- Maximum Modulus Theorem (version one)
- Index of a closed curve
- Cauchy’s Theorem and Integral Formula (various versions)
- The Argument Principle (version 1)
- Open Mapping Theorem
- Singularities
- Types of singularities
- Order of poles and local form of function
- Laurent series
- Casorati-Weirstrass Theorem
- Residues and the Residue Theorem
- Contour integral calculations
- The Argument Principle (full version)
- Rouche’s Theorem
- The Maximum Modulus Theorem
- The Maximum Modulus Theorem (various versions)
- Schwarz’s Lemma
- Compactness and Convergence in the Space of Analytic Functions
- Spaces of continuous functions
- Normal families
- Equicontinuity
- Arzela-Ascoli Theorem
- Spaces of analytic functions
- Hurwitz’s Theorem
- Montel’s Theorem
- The Riemann Mapping Theorem
- Innite products
- The Weierstrass Factorization Theorem
- The gamma function
- The Riemann zeta function
- Harmonic Functions
- Mean value property
- Maximum principle (various versions)
- Harmonic functions on a disk
- The Poisson kernel
- Harnack’s Theorem
- Subharmonic and superharmonic functions
- Perron families
- Goursat’s Theorem
- Convex functions and Hadamard’s Three Circles Theorem
- The Phragmen-Lindelof Theorem
- Spaces of meromorphic functions
- Runge’s Theorem
- Mittag-Leer’s Theorem
- Schwarz Reflection Principle
- Analytic Continuation along a path
- The Dirichlet Problem
- Green’s Function
- Entire functions
- Little and Great Picard Theorems.
Additional topics: These might be included depending on who taught your course and/or is making up the exam. Be sure to check.
(posted March 26, 2014)