Math 245: Introduction to Complex Analysis

Course Goals:

The course is intended to introduce first year graduate students to complex analysis. The topics to be covered will match very closely the topics on the syllabus for the qualifying exam in complex analysis. These topics are posted at http://www.math.duke.edu/graduate/qual/qualcompanal.html. Strong undergraduates may find this course a suitable introduction to the subject. In comparison with the undergraduate complex analysis course, math 181, this course will be probably be faster paced, cover more material, and pay more attention to rigorous argument.

Intended audience:

This is an important basic course for first year graduate students irrespective of whether they intend to specialize in pure or applied mathematics. It is a prerequisite for a variety of second year graduate courses.

Prerequisites:

Math 204 or equivalent.

Text:

Functions of one complex variable, I, second edition, by John B. Conway.

There will be weekly homework assignments. There may also be an examination.

Weekly homework assignments:

1. Homework for Tuesday, January 14.

• Please read in Conway sections 1.1 - 1.4 and work the following problems:
• 1.1 problems 1 and 2.
• 1.4 problems 1, 2, and 7.
• Please read in Conway the material on the Cauchy-Riemann equations (bottom of page 40 through 2.29).
• Please work the problems on the handout, exercise set 1.

2. Homework for Tuesday, January 21.

• Please work the following problems in Conway:
• Section 2.6 First sentence of exercise 1. (You may ignore the rest of exercise 1.)
• Section 3.1 4, 5, 6 (a), (b), (d), 7.
• Problem 2 on exercise set 1, if you did not work it last time.
• Exercise set 2 problems 1, 5, 6, 7, 8.

3. Homework for Tuesday, January 28.

• Please work the following problems in Conway:
• Section 3.2 4, 6, 7.
• Exercise set 2, problems 2,3,4.
• Exercise set 3.
• Problem 2 on Power Series, Generating Functions, Combinatorics.
• Remark: This assignment is too long.

4. Homework for Tuesday, February 4.

• Please work the following problems in Conway (when the word analytic
• appears use the definition given in class ie. everywhere locally given
• by a convergent power series.):
• Section 3.2 10,11,12(note conventions p. 40),13,14,19
• Exercise set 3 1/2.

5. Homework for Tuesday, February 11.

• Please skim section IV.1 and read sections IV.2 and IV.3 in Conway.
• Please work the following problems in Conway:
• Section IV.1 Problems 13, 19, 20.
• Section IV.2 Problems 1,5,6,7.
• Exercise set 4 problems 2,3,4,5.

6. Homework for Tuesday, February 18.

• Please work the following problems in Conway:
• Section IV.4 problem 4
• Section IV.5 problems 5, 6, 7
• Exercise set 5 problems 1, 2, 4.

7. Homework for Tuesday, February 25.

• Please read sections V.1 and V.2 through example 2.5 in Conway.
• Section V.1 Problems 1a,b,c,d,e,g,h; 4,5.
• Section V.2 Problems 1a; 2a, 3, 4,5.
• Exercise set 5 problem 5.

8. Homework for Tuesday, March 4.

• Section V.2 Problems 6, 7, 8, 9.
• Section V.3 Problems 1, 2,3.
• Exercise set 6 problems 1-5.

9. Homework for Tuesday, March 18.

• Please read sections IV.7, III.3 (except for pages 50-51, which may be left to the next assignment) and pages 252-253 in Conway.
• Exercise set 7.
• Exercise set 8.

10. Homework for Tuesday, March 25.

• Please skim sections I.5, I.6, VI.1 and read pages 50-51 in Conway.
• Exercise set 9. Exercise set 10 problems 0-4.

11. Homework for Tuesday, April 1.

• Exercise set 10 problems 5, 6.
• Exercise set 11 problems 1, 2.
• Exercise set 12 problem 2 parts (i)-(iii)

12. Homework for Tuesday, April 8.

• Please review Conway II.4 as needed. The material discussed this
• week in class is covered in Conway VII.1, VII.2, VII.4. At the
• moment, it is difficult to tell if we will be able to cover all
• this material in Thursday's lecture.
• Please complete exercise set 12.
• Please work in Conway VII.2 problem 1,
• VII.4 problems 2,4,9. (In the first two problems the task is
• to prove uniform convergence on compact sets.)

13. Homework for Tuesday, April 15.

• The Riemann mapping theorem is covered in Conway in VII.1, VII.2, VII.4.
• The treatment in Conway is largely the same as in class, with minor
• variations in detail and order of exposition.
• The topic of Thursday's lecture is the Weierstrass factorization thm.
• This is covered in Conway in VII.5. Due to time constraints we will
• not cover material beyond Thm. VII.5.14 in Conway.
• Please work exercise set 11 problems 3 and 4.
• Please work Conway VII.5.7 and VII.5.12.
• Please work exercise set 13 1,2, 7(i).

14. Homework for Tuesday, April 22.

• us to postphone a more in-depth discussion of the Riemann zeta function
• to an analyitc number theory class.
• The material on the gamma function is in Conway VII.7, although his
• presentation is organized a bit differently than the lecture. Conway VII.6
• is also related. The material on the Mittag-Leffler theorem is in
• Conway VIII.3. He proves a stronger version than proved in class
• (the domain is not assumed to be all of C), but bases his proof on
• Runge's theorem, which we will not cover.
• Please work exercise set 13 problems 3,4,5,6.
• Please work Conway VIII.3 exercise 1. (Assume that the domain G=the
• complex numbers.)

15. Analytic Continuation.

• Unfortunately, lack of time prevents us from covering the Schwarz reflection
• principle (Conway IX.1). Conway's treatment of analytic continuation is
• thorough, but not brief (Conway IX.2 - IX.7). Some may find the treatment
• in Ahlfors, Complex analysis, better, because it is briefer. Perhaps better
• still for those with some familiarity with manifolds is the treatment in
• Forster, Lectures on Riemann Surfaces, 4.1-4.10 and sections 6 and 7.