Mathematics 6321       Complex Analysis      Course Description

Class times: Tues-Thur. 1:30-3:00 in room 101 of the ESM Building.

Instructor: Evans Harrell, Skiles 134, tel. 894-9203, e-mail harrell@math.gatech.edu

Instructor's office periods: Tues.-Thur., beginning at 3:00.

Text: Stein and Shakarchi, Complex Analysis

Prerequisites: Students should be comfortable with analysis at the level of Math 4317 and the field of complex numbers, and acquainted with complex-valued functions at the level of Math 4320, or at least serious practical experience with complex-function theory. We shall begin Math 6321 with a serious discussion of most of the topics of Math 4320, but the material will be assumed familiar.

Grading and requirements: There will be exams on

and there may be pop quizzes on other Thursdays, when homework is due. It is possible that an additional full-fledged test or major assignment will be scheduled in April. Homework problems will be posed in the lectures or on the Web, and will be collected and reviewed but not systematically graded. They might, however, appear on the exams. You will have an opportunity to review your class standing from the Web page.

Missed tests, special accommodation, etc. Special accommodations are strongly discouraged. Any potential missed tests other than those due to documentable medical emergencies must be requested in writing or by electronic mail at least two weeks in advance of any scheduled event, and it is the student's responsibility to take the initiative for any such accommodations.

Description: Complex numbers have a mysterious origin. They are useful because, according to the fundamental theorem of algebra, any polynomial over the field of complex numbers has a root. Although the sixteenth-century algebraists did not know this theorem or understand much about the complex number field, their interest in polynomial equations gradually forced them to use and accept complex numbers. Even though they were seeking real roots of real polynomial equations - not at all guaranteed by a theorem - they could only succeed with complex methods. Much later, in the 1820's, Cauchy created the subject of complex analysis and thereby wonderfully connected algebraic, geometric, and analytic ideas. As we shall see, this is because of the unexpectedly deep consequences of differentiability for functions of a complex variable.

The theory of functions of a complex variable will have the classical look and feel of the nineteenth-century masterpiece that it is, but it remains a living subject both for its theory and applications. The topics we shall discuss are much as described in the School of Mathematics Course Description, but, depending on the interests and cohesiveness of the class, we may explore some alternatives.

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