Math 618-600 -- Theory of Functions of One Complex Variable II

General Description

This is a second semester rigorous course in the theory of functions of one complex variable. Topics include Riemann Mapping Theorem, Runge's Approximation Theorem and related topics. A more complete outline is given below.

Textbook

The required textbook is Function Theory of One Complex Variable by Robert E. Greene and Steven G. Krantz, Wiley, 1997. We will cover parts of chapters 6-12. Another useful reference is Real and Complex Analysis , by Rudin.

Prerequisite

The official prerequisite for this course is Math 617 Complex Analysis I. Math 617 and its successor Math 618 form the basis for the Mathematics Department Qualifying Examination in Complex Analysis.

Class Times and Office Hours

The course meets Tuesday and Thursday, 9:35-10:50am in BLOC 163. Office Hours are Monday thru Thursday 1:30-2:30pm. in BLOC 623. If those times are inconvenient, you can set up an appointment by e-mailing me (boggess@math.tamu.edu) or by calling 845-3261.

Grading

Course grades will be determined by homework (50%); a written project over some topic relating to the course material. You may collaborate with other students on the homework.

Course Outline

I. Riemann Mapping Theorem

Topology of uniform convergence on compact sets

Arzela-Ascoli Theorem

Normal Families

Proof of the Riemann Mapping Theorem

II. Infinite Products

Definitions and basic facts

Weierstrass Factorization Theorem

Mittag Leffler Theorem

Examples: the Gamma Function

III. Runge's Approximation Theorem and Related Consequences

Approximating a continuous function by a polynomial

Two proofs of Runge's Theorem (classical pole pushing and Hahn Banach)

The solution of the inhomogeneous Cauchy-Riemann equations

Mittag Leffler and Weierstrass Theorems (revisited)

Special Topics (as time permits)

Jensen's Formula

Hadammard's Factoriztion Theorem

Picard's Theorem

Complex Analytic Manifolds

H^p Theory