### MATH 6321 Complex Analysis

### Spring 2019

### TR 1:30-2:45 Skiles 256

### Office Hours: TR 4:30-5:30 in Skiles 133b

If you have problem to meet with me during the scheduled time, email me
to set an appointment.

**Syllabus**
See online syllabus at MATH
6321. We will cover the first six chapter of the texbook. This
will leave us sometime at the end of the course for topic that may be of interest to the
students.

#### Prerequisite

MATH 4317, 4320
#### Textbook

John B. Conway, "Functions of one complex variables I", Springer, 2nd
Edition.

#### Grading

I will collect HW every 2 weeks for a total of 5/6 HW. On top there
will be two in class midterms and a in class final. he first midterm
is tentatively scheduled for **Thursday February 21** and the
second on **Thursday March 28**. Midterms and final
will be cumulative. The possibility of personal project will be
discussed later during the class. The grading scheme will be 15% HW,
40% midterms and 45% final.
**First week**

- Material covered
- I.1 to I.6: The complex Number
- II.1 to II.6: Metric Spaces and the Topology of C

- Exercises
- I.6: 4
- II.2: 4
- II.4: 5
- II.5: 4, 9
- II.6: 1

**Second week**

- Material covered
- III.1: Power series
- III.2: Analytic function

- Exercises
- III.1: 6, 7
- III.2: 2, 6, 15

**Third week**

- Material covered

- III.3: Analityc Function as mapping. Möbius
transformations.
- IV.1: Riemann-Stiltjes integral.

- Exercises

- III.3: 9, 18
- IV.1: 13, 21,23

**Fourth week**
- Material covered

- IV.2: Power Series reppresentation of
analytic functions.
- IV.3: Zeros of analytic functions.

- Exercises

- IV.2: 4, 8, 10, 11
- IV.3: 8

**Fifth week**
- Material covered

- IV.4: The index of a closed curve.
- IV.5: Cauchy's Theorem and Integral Formula

- Exercises

The first midterm will be on Tuesday Febraury 21. Solution set for first midterm of Spring 2009 and Spring 13 for midterm 1.

**Sixth week**
- Material covered

- IV.6: The homotopic version of Cauchy's Theorem and
simple connectivity.
- IV.7: Counting zeros; the Open Mapping Theorem

- Exercises

- IV.6: 1, 4, 6, 10
- IV.7: 3, 4

**Seventh and Eighth week**
- Material covered:
- V.1: Classification of Singularities.
- V.2: Residues
- V.3: The Argument Principle

- Exercises:
- V.1: 1, 6, 10, 17
- V.2: 2 (c)(d)(h), 6, 7, 12
- V.3: 2, 7, 8

**Nineth week**
- Material covered:
- VI.1: The maximum Principle.
- VI.2: Schwarz's Lemma.

- Exercises:
- VI.1: 3, 5, 8
- VI.2: 2, 3, 5

**
The second medterm will be on Thursday March 28. ****Preparation material for the second midterm**. Second midterm with solution and final with solution from 2009.