MATH 6120 Complex Analysis

Textbooks

  • Elias Stein & Rami Shakarchi, Complex Analysis (in recent years)
  • Walter Rudin, Real & Complex Analysis (in previous years)

Minimum Syllabus

The topics in chapters 10–16 of Rudin are the absolute minimum.  The level of difficulty is about right for a basic course in the subject, but this does not mean that the course has to use this textbook. 

  1. Chapter 10 in Rudin.
  2. Cauchy-Riemann equation, mean value property, harmonic functions (chapter 11 in Rudin).
  3. Schwarz lemma, maximum modules theorem (chapter 12 in Rudin).
  4. Runge’s approximation theorem (chapter 13 in Rudin).
  5. Conformal mapping, normal families of holomorphic functions, Riemann mapping theorem (chapter 14 in Rudin).
  6. Mittag-Leffler theorem, Weierstrass theorem in existence of functions with prescribed zeroes (chapter 15 in Rudin).
  7. Analytic continuation (chapter 16 in Rudin).

Optional Topics

Depending on the instructor, different optional topics are covered.

  1. The equation ∂/ ∂{\bar z} = g.
  2. Riemann surfaces (notes by C. Earle).
  3. Distribution theory (textbook by Strichartz).
  4. Several complex variables (Strichartz, Hubbard).
  5. Prime number theorem (Hubbard).
  6. Introduction to complex dynamics (Hubbard).
  7. Uniformization theorem (Hubbard).