# Complex Variables 27000 Winter 2017

### Description of course:

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

### Office hours:

Calegari's Office hours are 1:30-2:20 Mondays.

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### 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 Friday and due before the start of class the following Friday. 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 Friday, January 13: 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 Friday, January 20: 1.4:5, 1.4:11, 1.4:15, 1.5:3, 1.5:7, 1.5:8, 1.5:13, 1.5:24, 1.6:8
• Homework 3, due Friday, January 27: 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
• Homework 4, due Friday, February 3: 2.2:11, 2.3:1, 2.3:2, 2.3:4, 2.3:7, 2.4:2, 2.4:3, 2.4:10
• Midterm, due Friday, February 10: here
• Homework 5, due Friday, February 17: 3.1:4, 3.1:12, 3.1:17, 3.2:5, 3.2:9, 3.2:22, 3.2:25, 3.3:3, 3.3:11
• Homework 6, due Friday, February 24: 4.1:2, 4.1:5, 4.1:12, 4.2:7, 4.2:8, 4.2:13, 4.3:3, 4.3:6, 4.3:10, 4.4:2
• Homework 7, due Friday, March 3: 5.1:4, 5.1:9, 5.2:3, 5.2:9, 5.2:20, 5.2:28, 5.2:31, 5.2:34
• Homework 8, due Friday, March 10: 6.1:2, 6.1:5, 6.1:7, 6.1:9, 6.1:11, 6.1:13
• Final, due Friday, March 17: here

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### 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.

1. review of complex numbers and their basic properties
2. complex functions; continuity, differentiability and the Cauchy-Riemann equations and the Laplace equation
3. power series; term by term differentiation, radius of convergence, etc.
4. line integrals and the Cauchy-Goursat theorem for a triangle
5. primitives and path independence; Cauchy's Thm. in a convex region
6. Cauchy integral formula, infinite differentiability of holomorphic functions; Cauchy estimates Liouville's Theorem and the Fundamental Thm. of Algebra, Morera's Thm.
7. power series representation of holomorphic functions
8. Laurent expansions of analytic functions in an annulus
9. residues and the residue theore
10. calculation of integrals using the residue theorem
11. sequences of analytic functions and almost uniform convergence
12. infinite products of analytic functions; Weierstrass theorem on entire functions with prescribed zeros
13. Cauchy's theorem for simply connected regions
14. maximum modulus principle
15. 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.