*Complex Variables 27000* Fall 2017

### Instructor: Danny Calegari; Grader: Shuo Pang

### TuTh 11:00-12:20 Eckhart 206

### Description of course:

This course is an introduction to Complex Variables. It is intended for
undergraduates.

### Office hours:

Calegari's Office hours are 12:30-1:20 Mondays.

### Cancellations:

None yet.

### Notices:

None yet.

### Homework/Midterm/Final

There will be a midterm and a final. There will also be weekly homework.

Grading is based 50% on homework, and 25% on each exam.

Homework is posted to this website each Thursday and due before the *start*
of class the following Thursday. Late homework will not be accepted.

Homework is usually taken directly from Marsden-Hoffman. The notation x:y
means exercise y from section x.

- Homework 1, due Thursday, October 5: 1.1:2, 1.1:7, 1.1:19, 1.1:20, 1.2:4, 1.2:9, 1.2:14, 1.2:25, 1.3:7, 1.3:19, 1.3:26, 1.3:33
- Homework 2, due Thursday, October 12: 1.4:5, 1.4:11, 1.4:15, 1.5:3, 1.5:7, 1.5:8, 1.5:13, 1.5:19, 1.6:8
- Homework 3, due Thursday, October 19: 2.1:1, 2.1:4, 2.1:5, 2.1:8, 2.1:10, 2.1:15, 2.1:16, 2.2:5, 2.2:8

### Notes:

None yet.

### Syllabus:

The draft syllabus is as follows; note that this is based on a previous
lecturer's syllabus, and is subject to revision, especially towards the
later selection of topics.

- review of complex numbers and their basic properties
- complex functions; continuity, differentiability
and the Cauchy-Riemann equations and the Laplace equation
- power series; term by term differentiation, radius of convergence, etc.
- line integrals and the Cauchy-Goursat theorem for a triangle
- primitives and path independence; Cauchy's Thm. in a convex region
- Cauchy integral formula, infinite differentiability
of holomorphic functions; Cauchy estimates Liouville's Theorem
and the Fundamental Thm. of Algebra, Morera's Thm.
- power series representation of holomorphic functions
- Laurent expansions of analytic functions in an annulus
- residues and the residue theore
- calculation of integrals using the residue theorem
- sequences of analytic functions and almost uniform convergence
- infinite products of analytic functions; Weierstrass theorem
on entire functions with prescribed zeros
- Cauchy's theorem for simply connected regions
- maximum modulus principle
- the argument principle

### References:

The main reference is Basic Complex Analysis, Third Edition, by
Jerrold Marsden and Michael Hoffman. Homework problems will largely be taken
from this book, so it is important to have access to it. Lectures may diverge
from the book both in style and topics.