Complex Analysis

Complex Analysis
Introduction to theory of complex analysis at beginning graduate level. Topics: Cauchy integral equations, Riemann surfaces, Picard's theorem, etc.
MATH
532
 Hours3.0 Credit, 3.0 Lecture, 0.0 Lab
 PrerequisitesMath 352 or instructor's consent.
 TaughtFall Contact Department, Winter Contact Department, Spring Contact Department, Summer Contact Department
 ProgramsContaining MATH 532
Course Outcomes

Learning Outcomes

Students should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the concepts below, related to, but not identical to, statements proven by the text or instructor.

Overview

Essential results: Power series, integration along curves, Goursat theorem, Cauchy's theorem in a disc, Taylor series, Morera's theorem, singularities, residue calculus, Laurent series, argument principle, harmonic functions, maximum modulus principle.

Entire functions: Jensen's formula, functions of finite order, Weierstrass infinite products, Hadamard factorization theorem.

The gamma and zeta functions: Analytic continuation of gamma function, further properties of Γ, functional equation and analytic continuation of zeta function.

Conformal mappings: Conformal equivalence, Schwarz lemma, Montel's theorem, Riemann mapping theorem.

Elliptic Functions: Liouville's Theorems, poles and zeros of elliptic functions, Weierstrass elliptic functions.

For more detailed information visit th Math 532 Wiki page.