Math 5410 -- Functions of a Complex Variable

Lectures: TR 3:30 p.m. - 4:50 p.m. in Curry 210.

Instructor: William Cherry

Office Location:	General Academic Building (GAB) 405.
E-mail:	wcherry@unt.edu

Prerequisites

Although the course has no formal prerequisites, and in particular no previous knowledge of complex numbers or complex functions will be assumed, students should have had some prior exposure to a proof based analysis or calculus course on the level of Math 3610. In particular, students should have had some exposure to uniform convergence and a proof of Taylor's Theorem.

Alternative Courses

Advanced undergraduate students who have taken a course such as Math 3610, particularly those students planning to do graduate work in mathematics or the sciences, are welcome to enroll in Math 5410. However, undergraduate students with no prior exposure to complex functions may want to consider Math 4520, the undergraduate course, as an alternative. Math 4520 puts less of an emphasis on proofs of the major theorems in complex analysis and puts more emphasis on what complex analysis can be used for.

Course Description

The course will cover the basics of functions of one complex variable. The emphasis on the course will be proofs of the foundations of complex analysis. However, there will also be some discussion of applications to fields such as electrostatics and fluid dynamics. The course is also aimed at preparing mathematics graduate students for the departmental exam in complex analysis. Topics will include: an introduction to the complex numbers, polar coordinates, stereographic projection, complex functions defined as power series, the elementary transcendental functions, fractional linear transformations, complex derivatives, the Cauchy-Riemann equations, complex integration, the Cauchy Integral Formula and Cauchy's Theorem, the Calculus of Residues, conformal mappings and their applications, and perhaps some discussion of harmonic functions. Geometric intuition is a very important part of complex function theory, and topics will be presented from the geometric viewpoint whenever possible. The section on Cauchy's Theorem will include an introduction to the topology of the plane, but no prior knowledge in topology will be assumed.

Textbook

Bruce Palka, An Introduction to Complex Function Theory, Springer-Verlag, 1995

Course Requirements

The course will have two "in-class" mid-term exams, a "take-home" mid-term exam, and a final exam which will have both a "take-home" and "in class" component. Students may also be asked to occasionally present solutions to homework problems. The relative weight of each of these in determining final grades wil be:

Homework	10%
In Class Midterms	20% (10% each)
Take Home Midterm	15%
In Class Final Exam	25%
Take Home Final Exam	30%

Important Dates

Final Exam (in class)	Thursday, December 11, 1:30-3:30
Final Exam (take home)	Due, Thursday, December 11, 1:30
In Class Midterms	Tuesday, September 30 and Tuesday, November 18
Take Home Midterm	Due, Tuesday, October 21

Homework Assignments, etc.

Coruse Syllabus
Homework due 8/28
Homework due 9/4
Homework due 9/11
Homework due 9/18
Homework due 9/25 and 10/2
Homework due 10/9

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