**Math
416, Fall 2017 **

**Complex
Analysis **

**Instructor
**John E. M^{c}Carthy

**Class **
TuTh
11.30-1.00 in Eads 204

**JM Office **
105 Cupples I

**JM Office Hours** M 3.00-4.00,
Tu 2.00-3.45, Th
10.00-11.30, and by appointment

**Phone **
935-6753

**Exams** There will be two exams in the course:

1) Exam 1 In
class. Thursday October 19.

2) Exam 2 Final exam. Monday December
18, 1.00-3.00.

**Homework**

There will be weekly homework sets during the semester, assigned on Tuesday and due the following Tuesday.

Homework 1, due September 5.

Homework 2, due September 12.

Homework 3, due September 19.

Homework 4, due September 28.

Homework 5, due October 3.

Homework 6, due October 10.

Homework 7, due October 24.

**Prerequisites**

Math 318, or permission of instructor.

**Description**

Complex Analysis is an essential tool in (almost) all areas of modern
mathematics. It started with

Tartaglia's solution of the cubic - in order to find the real roots of a
real cubic polynomial, the formula

requires complex numbers. The fundamental theorem of algebra says that
every complex polynomial

can be factored into linear factors. This means that every matrix has
complex eigenvalues, though not necessarily real ones.

Analytic functions of complex variables - functions that can locally be
written as power series - are the heart of the subject.

They are both flexible and rigid, in ways we will discuss, and make the
subject very attractive.

**Content**

- Complex Numbers. De Moivre's Formula
- Complex Differentiation. Cauchy-Riemann equations.
- Harmonic Functions
- Linear fractional transformations
- Exponential and logarithmic functions
- Power series
- Complex Integration
- Cauchy's theorem
- Some of the many consequences of Cauchy's theorem - Liouville's theorem, Maximum modulus theorem, Schwarz's lemma
- Harmonic functions redux
- Laurent series. Singularities and Poles.
- Residue theorem. Definite integrals.
- Rouche's theorem. Riemann mapping theorem.
- Homotopy version of Cauchy's theorem.

Basis for Grading

Attendance and class participation will be 5% of your grade, homework
will be 30%, the midterm will be 30%, and the final will be 40%.

**Homework**

Homework is an extremely important part of the course. Whilst talking
to other people about it is not dis-allowed, too often this degenerates
into one person solving the problem, and other people copying them
(often justified to themselves by saying "I provide the ideas, X does
the details" - but the details are the key. If you can't translate the
idea into a real proof, you don't understand the material well enough).
So I shall introduce the following rules:

(a) You can only talk to some-one else about
a problem if you have made a genuine effort to solve it yourself.

(b) You must write up the solutions on your own. Suspiciously similar
write-ups will receive 0 points.

**Class**

I expect you to come to class every day, and to participate in
class discussions.

I also expect you to stay abreast of the material we are covering, and
may call on you at any time to answer a question.

Class etiquette: don't be disruptive or discourteous. No beeping, ringing, crunching, rustling, leaving early or arriving late. No texting, sleeping, checking your phone.

**Texts **
*Complex Function Theory* by Donald Sarason (AMS,
2007)

**Additional Reading
**

Any book on complex analysis in Olin library will contain all the
material we cover.

Find one whose style you like and check it out.