Last update on 4/19/02.

MAT 542 Spring 2002

Course Instructor: Irwin Kra, 4-111 Math Tower, 2-8273, irwin@math.sunysb.edu. Course meets: M, F 12:40-2:00pm in Physics P-124. The first meeting is on 1/25. IK office hours: W 3-4, Th 3-4 in 4-111, F 11-12 in the Undergraduate office, and by appointment. Course assistant/grader: Yasuhiro Tanaka, 2-117 tanaka@math.sunysb.edu. YT office hrs: Tu 1:20-2:20pm.

The course will cover the basic syllabus described in the graduate student handbook. I would like to discuss, in addition, the Prime Number Theorem as an important application of complex analysis to number theory. If time permits (unlikely), we will provide an introduction to one or more of the following topics depending on student interests:

1. Elliptic function theory.

2. Algebraic functions.

3. Subharmonic functions.

The first lecture is on Friday, January 25. Homework will be assigned, collected and graded. Problems will vary from routine ones to the very challenging level. There will be one midterm examination held on Friday, March 15 in class and a final examination on Friday, May 10, 11-1:30pm in a room to be announced.

GRADES will be calculated using the following scheme: Homework 20%; Midterm 20%; Final 60%. In computing the homework grade, the one with the lowest score will be dropped.

If you have any condition, such as a physical or mental disability, that will make it difficult to complete the course or that will require extra time on examinations, please notify me as soon as possible so that appropriate arrangements can be made.

There are many fine books on the subject. Although I will not follow anyone completely, reading one will greatly increase your understanding. Some of the books I am familiar with are:

1. L.V. Ahlfors, Complex Analysis (third edition).

2. R.P. Boas, Invitation to Complex Analysis.

3. C. Carathéodory, Theory of Functions.

4. J.B. Conway, Functions of One Complex Variable (second edition).

5. R.E. Greene and S.G. Krantz, Function Theory of One Complex Variable.

6. E. Hille, Analytic Function Theory.

7. R. Narasimhan, Complex Analysis in One Variable.

8. W. Rudin, Real and Complex Analysis.

Some of these books have been placed on reserve in the library.

I plan to present a relatively self contained treatment of complex function theory; the necessary foundations from algebra, real analysis and topology will be reviewed (mostly in the first lecture). This announcement and its updates will be posted on the web (there will be a pointer to it in my home page (www.math.sunysb.edu/ĩrwin).

Lectures and Examinations

L1: 1/25/02. Foundations, notations and background. L2: 1/28. Uniform continuity, (complex) derivatives, Cauchy-Riemann equations. L3: 2/1. Complex power series, computations of radius of convergence. L4: 2/4. Properties of power series, Abel’s limit theorem, exponentials and related functions. L5: 2/8 Periodicity of the exponential function, logarithms. L6: 2/11 Functions with power series expansions, zeros and poles of meromorphic functions. L7: 2/15 Analytic continuation. L8: 2/18 Line integrals. L9: 2/22 Closed and holomorphic 1-forms. L10: 2/25 The winding number of a curve. L11: 3/1 The Cauchy theory. L12: 3/4 Cauchy’s integral formula and consequences. L13: 3/8 Liouville’s theorem and consequences. Mean value property and maximum modulus principle. Schwarz’s lemma. Laurent series. L14: 3/11 Classification of isolated singularities. The residue theorem. Midterm: 3/15. L15: 3/18 Evaluation of (real) definite integrals. L16: 3/22 Meromorphic functions. L17: 4/1 Sequences and series of holomorphic functions. Metrics on C(D) and H(D). Example of partial fraction expansions. Convergence of series of meromorphic functions. L18: 4/5 Compactness in H(D). Conformal equivalence. Local properties of holomorphic maps. L19: 4/8 Möbius trasnsformations. The group of conformal self maps of D (for select D). L20: 4/12 The Riemann mapping theorem. L21: 4/15 Harmonic functions and the Dirichlet problem. L22: 4/19 Harmonic functions and the Poisson formula. L23: 4/22 Zeros of holomorphic functions and infinite products. L24: 4/24 Weierstrass theorems. Special Lecture: 4/26 Popescu on “Hartog’s phenomenon.” L25: 4/29 The Gamma function. L26: 5/3 The Prime Number Theorem. L27: 5/6 The Prime Number Theorem (conclusion). Final examination: 11-1:30pm, F 5/10.

Homework Assignments

Number Date posted Date due
1  1/28  2/8
2  1/31  2/15
3  2/4  2/22
4  2/5  3/4
5  2/22  3/11
6  2/25  3/18
7  3/6  4/5
8  3/15  4/12
9  3/15  4/19
10  4/24  4/29