Stephen P. Shipman
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Department of Mathematics
Louisiana State University
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Complex Analysis

Math 7350
Louisiana State University
Fall Semester, 2014

Prof. Stephen Shipman


Place: Lockett 111
Time: 11:30-12:20 Monday, Wednesday, Friday

Office: Room 314 of Lockett Hall
Telephone: 225/578-1674
Email: shipman@math.lsu.edu
Office Hours: Monday 1:30-4:00 and Thursday 1:30-3:30, or by appointment


Textbook

Complex Analysis by Elias Stein and Rami Shakarchi; Princeton Lectures in Analysis.

Course Description

Theory of holomorphic functions of one complex variable; path integrals, power series, singularities, mapping properties, normal families, special functions (Gamma, Zeta, Theta, Elliptic) Fourier transform, special topics.

Prerequisite

The prerequisite is a good graduate or very good undergraduate analysis course.

Course Topics

  1. Holomorphic functions and power series
  2. Cauchy's integral formula; Runge's approximation theorem
  3. Meromorphic functions and the logarithm
  4. The Fourier transform
  5. Infinite products and entire functions
  6. The gamma-function
  7. The Riemann zeta-function and the prime number theorem
  8. Conformal mappings--automorphisms of the disk, the Riemann mapping theorem, and polygons
  9. Elliptic functions
  10. Theta functions
  11. The Jordan curve theorem
  12. Some asymptotics

Assignments

The book has lots of exercises and problems; see the schedule of weekly problem sets below.

Students may discuss problems with each other and other people (including me, of course) and consult other literature; in fact students are encouraged to search the literature and discuss ideas. However, all work that is turned in must ultimately be that of the submitter alone. If a student receives aid on an assigned problem from discussions with people or other sources, he or she must begin from scratch in writing the solution so that the result is the product of his or her own understanding alone.

Evaluation

Regular assignments: 50%
Midterm exam: 20% Friday, October 17
Final exam: 30% Monday, December 8 from 10:00 to 12:00 in Lockett 111
The final exam will be based on the following themes:
Chapter 9, Exercise 2 (p. 278)
Chapter 9, Problem 2 (p. 281)
Chapter 4, Exercise 7 (p. 128)
Computing definite integrals by complex contour integration and residue calculus

Problems to do

Due date Section Problems to do
Fri., Sept. 5 Chapter 1 3, 7, 9, 13, 15, 16, 17, 18, 19, 23, 25
Mon., Sept 22 Chapter 2 Exercises 1, 4, 6, 9, 13, 15; Problem 4
Mon., Oct 6 Chapter 3 Exercises 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 22
. . . Chapter 4 None
Fri., Oct. 17 Midterm Exam Topics in residue calculus
Mon., Nov. 3 Chapter 5 Exercises 1, 2, 3, 5, 16, Problems 1, 2
. . . Chapter 6 None
. . . Chapter 7 None
Wed., Nov. 26 Chapter 8 Exercises 3, 13, 15; Problem 4

x@math.lsu.edu (x=shipman)