Last update on 3/25/99.

MAT 542 Spring 1999

Course Instructor: Irwin Kra 4-111, 2-8273, irwin@math.sunysb.edu. Course meets: Tu, Th 11:20-12:40pm in Physics P-130. The first meeting is on 1/21. IK office hours: Tu 2-3, W 11-12 (in Undergraduate Office), Th 10-11 and by appointment. Course assistant/grader: Rares Rasdeaconu 2-114, rares@math.sunysb.edu. RR office hrs: M 12-1.

The course will cover the basic syllabus described in the graduate student handbook. 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 Th January 21. Homework will be assigned, collected and graded. Problems should vary from routine ones to the very challenging level. There will be one midterm examination held on Th March 25 in class and a final examination on Tu May 11, 8:30-11:30am in Physics P-125.

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. Caratheodory, 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

L1: 1/21/99. Foundations, Notations and Background. L2: 1/26. Uniform continuity, (complex) derivative. L3: 1/28. Cauchy-Riemann, complex power series. L4: 2/2. Computations of radius of convergence. L5: 2/4. Abel’s limit theorem. Exponentials. L6: 2/9 Logarithms. More on power series. L7: 2/11 Power series. Zeros and poles of meromorphic functions. L8: 2/16 Line integrals. L9: 2/18 Holomorphic 1-forms. L10: 2/23 The winding number of a curve. L11: 2/25 Winding number (continued). The Cauchy theory. L12: 3/2 Cauchy’s integral formula and consequences. L13: 3/4 Extended form of Cauchy’s theorem. Liouville’s theorem and consequences. Mean value property and maximum modulus principle. L14: 3/9 Schwarz’s lemma. Laurent series. Classification of isolated singularities. L15: 3/11 The residue theorem. Evaluation of (real) definite integrals. L16: 3/16 Meromorphic functions. L17: 3/18 Sequences and series of holomorphic functions. L18: 3/23 Metric on C(D) and H(D). Midterm: 3/25 L19: 4/6 Convergence of series of meromorphic functions; partial fraction expansions; characterization of compact subsets of C(D). L20: 4/8 Compacteness in H(D). Conformal equivalence. L21: 4/13 Local properties of holomorphic maps. Möbius trasnsformations. The group of conformalself maps od D (for select D). L22: 4/15 The Riemann mapping theorem. L23: 4/20 Harmonic functions. L24: 4/22 Dirichlet problem and the Poisson formula. L25: 4/27 Zeros of holomorphic functions and infinite products. L26: 4/29 Weierstrass theorems. L27: 5/4 Weierstrass theorems (conclusion). Final examination: 8:30-11:30am, Tu 5/11 in Physics P-125.

Homework Assignments

Number Date handed out Date due
1  1/28  2/12
2  2/9  2/19
3  2/18  2/26
4  2/25  3/5
5  3/9  3/19
6  3/16  4/6
7  3/25  4/14
8  4/8  4/21
9  4/15  4/28