## Complex Analysis

Department:
MATH
Course Number:
6321
Hours - Lecture:
3
Hours - Lab:
0
Hours - Recitation:
0
Hours - Total Credit:
3
Typical Scheduling:
Every spring semester

Complex integration, including Goursat's theorem; classification of singularities, the argument principle, the maximum principle; Riemann Mapping theorem; analytic continuation and Riemann surfaces; range of an analytic function, including Picard's theorem.

Prerequisites:

MATH 4317 and MATH 4320 or equivalent

Course Text:

At the level of Conway, Functions of One Complex Variable

Topic Outline:
• Analytic functions
• Series and integration theorems and formulas; Goursat's theorem
• Singularities, the argument principle, Rouche's theorem
• Conformal mapping by elementary functions
• Harmonic families and Poisson's formula
• The maximum principle and Schwarz's lemma
• Spaces of analytic functions and normal families
• The Riemann mapping theorem and the Weierstrass factorization theorem
• Analytic continuation, multi-valued analytic functions, and Riemann surfaces
• Additional topics as time permits and interest dictates, e.g., the theorems of Runge, Picard, and Mittag-Leffler, Bergman's kernel, moment problems, elliptic functions, zeros of analytic functions, the Schwarz-Christoffel transformation