Math 220B - Complex Analysis

Welcome to Math 220B!

Course description:

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

Topics include: sequences, series and products of holomorphic functions, Weierstrass factorization, Mittag-Leffler theorem, normal families, Schwarz lemma and applications, Riemann mapping theorem, Schwarz reflection, Runge's theorem.

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

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

Office hours: M 3:30-5, APM 6-101, or by appointment

Teaching Assistant: Jonathan Aberle, jaberl "at" you-know-where dot edu

Office Hours: WF 1-3PM, APM 6-132.

Textbook: Functions of One Complex Variable, by J. B. Conway. In Math 220A, we will cover material from Chapters VI, VII, VIII.

Additional Reading: Complex Analysis, by Lars Ahlfors.

Prerequisites:

Math 220A. 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 February 12 as well as a Final Exam (in class) on Monday, March 18, 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 or Thursdays 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, January 8
University Holiday: Monday, January 15
Midterm Exam: Monday, February 12
University Holiday: Monday, February 19
Last class: Wednesday, March 13
Final Exam: Monday, March 18, 11:30-2:30 PM

Lecture Summaries

Lecture 1: Outline of the course and motivation. The Weierstrass and Mittag-Leffler problems. Infinite products. Turning convergent products into convergent series. Absolutely convergent products - PDF

Lecture 2: Products of holomorphic functions and their zeros. Logarithmic derivatives of infinite products. Factorization of the sine function - PDF

Lecture 3: Gamma function. Properties of the Gamma function. Identities, residues, Gauss' formula, integral representation - PDF

Lecture 4: The Weierstrass problem. Weierstrass elementary factors. Weierstrass factorization. Any meromorphic function in the complex plane is quotient of two entire functions - PDF

Lecture 5: The Mittag-Leffler problem in the complex plane - PDF
Lecture 6: Four historically important examples. The Weierstrass zeta and pe functions. Reinterpretation of Weierstrass in term of divisors and principal divisors. Introduction to Montel's theorem - PDF

Lecture 7: Normal families. Equicontinuity. Arzela-Ascoli theorem stated. Obtaining Montel's theorem from Arzela-Ascoli - PDF
Lecture 8: Proof of the Arzela-Ascoli theorem via Cantor diagonalization - PDF
Lecture 9: Mapping theory. Some guiding questions. Schwarz Lemma and its proof. Automorphisms of the unit disc - PDF
Lecture 10: Applications of Schwarz Lemma. Schwarz-Pick Lemma. Automorphisms of the punctured disc - PDF
Lecture 11: More on automorphism groups. Introduction to Riemann Mapping Theorem. Strategy for the proof - PDF
Lecture 12: Proof of Riemann Mapping Theorem. Remarks about the extremal function. Simply connected domains and taking roots - PDF
Lecture 13: Schwarz reflection principle. Applications to biholomorphisms of rectangles - PDF
Lecture 14: Introduction to Runge's theorem for compact sets. Polynomial and rational approximation. Polynomial approximation and converse to Little Runge - PDF
Lecture 15: Proof of Runge's theorem for compact sets. Cauchy's Integral formula for compact sets. Approximation by rational functions. Pole pushing to prescribed locations - PDF
Lecture 16: Runge's theorem for open sets and its proof. Examples - PDF
Lecture 17: Several topological and analytic characterizations of simply connected regions - PDF

Homework:

Homework 1 due Tuesday, January 16 - PDF. Solutions - PDF
Homework 2 due Tuesday, January 30 - PDF. Solutions - PDF
Homework 3 due Thursday, February 8 - PDF. Solutions - PDF
Homework 4 due Thursday, February 22 - PDF. Solutions - PDF
Homework 5 due Tuesday, March 5 - PDF. Solutions - PDF
Homework 6 due Tuesday, March 12 - PDF. Solutions - PDF

Preparation for Midterm and Final:

Midterm - PDF. Solutions PDF.
Final - PDF. Solutions PDF.
Midterm from Winter 2021 (modified) - 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 Winter 2021 - PDF. Solutions PDF
Dragos will hold office hours for the Final on Friday, March 15, 1 - 2:30 pm.