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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
- Holomorphic functions and power series
- Cauchy's integral formula; Runge's approximation theorem
- Meromorphic functions and the logarithm
- The Fourier transform
- Infinite products and entire functions
- The gamma-function
- The Riemann zeta-function and the prime number theorem
- Conformal mappings--automorphisms of the disk, the Riemann mapping theorem, and polygons
- Elliptic functions
- Theta functions
- The Jordan curve theorem
- 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
|
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