Math 220A - Complex Analysis

Welcome to Math 220a!

Course description:

This is the first in a three-sequence graduate course on complex analysis.

Topics include: holomorphic and meromorphic functions, Taylor and Laurent expansions, Cauchy's theorem and its applications, calculus of residues, the argument principle, sequences of holomorphic and meromorphic functions, and others.

Instructor: Dragos Oprea, doprea "at" you-know-where dot edu

Lectures: MW 11:00-12:20, APM B412

Office hours: M 1:00-2:30, APM 6-101, or by appointment

Teaching Assistant: Robert Koirala, rkoirala "at" you-know-where dot edu

Office Hours: W 3-5, HSS 4059

Textbook: Functions of One Complex Variable, by J. B. Conway. In Math 220A, we will cover material from Chapters III, IV, V and a couple of sections from Chapters VI, VII.

Additional Reading: Complex Analysis, by Lars Ahlfors.

Prerequisites:

Math 140AB. However, this is a graduate level course, so at times, we may use notions from related fields, including topology and real analysis. I am happy to discuss prerequisites on an individual basis. If you are unsure, please don't hesitate to contact me.

The syllabus can be found here.

Grading:

The final grade is based on homework (30%), midterm (30%) and final exam (40%).

There will be a Midterm (in class) on Monday, November 6 as well as a Final Exam (in class) on Tuesday, December 12, 11:30-2:30PM.

The problem sets are mandatory and are a very important part of the course. The problem sets are due on Tuesdays at 5pm on Gradescope. There will be a 24hr grace period, but no other extensions will be given. We will drop the lowest homework.

There are several ways to upload your homework on Gradescope. A possible method is described here.

Working with your peers is acceptable, but solutions must be written independently.

Academic Integrity

Any violation of the Academic Integrity Policies is considered a serious offence at UCSD. Students caught cheating will face administrative sanctions. No credit will be given on the assignment/exam which resulted in the academic integrity violation.

Important dates:

First class: Monday, October 2
Midterm Exam: Monday, November 6
Veterans Day: Friday, November 10
Thanksgiving: Thursday-Friday, November 23-24
Last class: Wednesday, December 6
Final Exam: Tuesday, December 12, 11:30-2:30 PM

Lecture Summaries

Lecture 1: Complex differentiable functions. Comparison with real differentiability. Cauchy-Riemann equations. Harmonic functions - PDF

Lecture 2: Power series. Radius of convergence. Analytic functions. Differentiability of power series. Logarithm - PDF

Lecture 3: Conformal maps. Biholomorphisms. Mobius transformations and generalized circles. Triple transitive action on the extended plane. Cayley transform - PDF

Lecture 4: Integration over continuously differentiable paths. Basic properties and examples. Existence of primitives over arbitrary regions and over discs - PDF

Lecture 5: Proof of Goursat's lemma. Corollaries. Cauchy's integral formula. Winding number - PDF

Lecture 6: Properties of winding numbers. Homotopic loops. Homotopy form of Cauchy's theorem. Homotopy form of Cauchy's integral formula. Primitives and logarithm in simply connected sets - PDF

Lecture 7: Proof of homotopy version of Cauchy's theorem. Taylor's theorem. Cauchy's estimates - PDF

Lecture 8: Homotopy form of Cauchy's integral formula for derivatives. Liouville's theorem. Fundamental theorem of algebra. Zeros of holomorphic functions. The identity principle - PDF

Lecture 9: Open mapping theorem. Maximum modulus principle. Laurent series. Types of singularities - PDF

Lecture 10: Characterization of singularities. Removable singularity theorem. Poles. Essential singularities. Casaorati-Weierstrass - PDF

Lecture 11: Methods of computing residues. Statement of the residue theorem. Examples and special cases. Nullhomologous chains and connection with nullhomotpic chains. Enhancemments of Cauchy's theorem and CIF - PDF

Lecture 12: Proof of CIF and Cauchy's theorem for nullhomologous chains. The proof of the residue theorem - PDF

Lecture 13: Applications of the residue theorem. Trigonometric integrals, Integrals of rational functions. Fourier integrals - PDF

Lecture 14: Applications of the residue theorem. Logarithmic integrals. Mellin integrals - PDF

Lecture 15: Singularities and residues at infinity. Residue theorem on the extended complex plane. Applications - PDF

Lecture 16: The argument principle and generalizations. Applications: zeroes and poles of elliptic functions - PDF

Lecture 17: Biholomorphisms and integral formula for the inverse. Rouche's theorem and applications - PDF

Lecture 18: Types of convergence for holomorphic functions. Local uniform convergence and uniform convergence on compact sets. Series of holomorphic functions - PDF

Lecture 19: Hurwitz's theorem and consequences. Preview of Math 220B - PDF

Homework:

Homework 1 due Tuesday, October 10 - PDF. Solutions - PDF
Homework 2 due Tuesday, October 17 - PDF. Solutions - PDF
Homework 3 due Tuesday, October 24 - PDF. Solutions - PDF.
Homework 4 due Tuesday, October 31 - PDF. Solutions - PDF
Homework 5 due Tuesday, November 14 - PDF. Solutions - PDF
Homework 6 due Tuesday, November 21 - PDF. Solutions - PDF
Homework 7 due Tuesday, November 28 - PDF. Solutions - PDF
Homework 8 due Tuesday, December 5 - PDF. Solutions - PDF

Preparation for Midterm and Final:

Midterm - PDF. Solutions PDF.
Practice Problems for Midterm - PDF.
Midterm from Fall 2020 - PDF. Solutions PDF.
Additional Practice Problems for Final - PDF. There is no need to solve all the problems on this list; there are more problems than you may possibly need, so just try a few that you like.
Final from Fall 2020 - PDF. Solutions PDF
Office Hours for the final: Dec 11, 1-2:30pm.
Final - PDF. Solutions PDF