## Math 245: Introduction to Complex Analysis

Instructor: Chad Schoen

### 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.
### Grading:

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 read in Conway sections 2.6 and 3.1. Please also read pages 33-38.
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 read pages 33-38 in Conway.
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 read pages 39-43 in Conway.
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 read sections IV.4, IV.5 and IV.8 in Conway.
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.

Please read section V.3 in Conway.
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.

Please read Conway VI.2 and IX.2.1 - IX.2.13.
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.

Please read Conway VII.8.1 and VII.8.17-19. Time constraints force
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.
Last modified April 14, 2003