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, normal families, Schwarz lemma and applications, Riemann mapping theorem, Weierstrass factorization, Runge's theorem, Mittag-Leffler theorem, and others.

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

Online lectures: MWF 3:00-3:50

Virtual office hours: W 4:00 - 5:30 (tentatively)

Teaching Assistant: Shubham Sinha, shs074 "at" ucsd

Virtual office Hours: Thursdays, 4-6 PM

Online teaching:

We will use Zoom for both lectures and virtual office hours. Regular attendance is expected (barring unforeseen circumstances and special arrangements). You can access the Zoom Room for this course from Canvas by cliking on "Zoom LTI PRO" at the bottom of the menu on the left. The Zoom Meeting ID is available via Canvas.

The lectures will be given synchronously. Class sessions will be recorded and later made available on Canvas. Sharing the recordings or the links to the recordings with anyone is prohibited.

Textbook: Functions of One Complex Variable, by J. B. Conway.

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.

Grading:

There will be a take home Midterm on Friday, February 12, as well as a Final Exam on Wednesday, March 17, 3-6PM.

The final grade is based on homework (30%), midterm (30%) and final exam (40%). The problem sets are mandatory and are a very important part of the course.

The problem sets are due on Fridays on Gradescope. In order not to interfere with lecture time, there will be a grace period on gradescope.

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

Important dates:

First class: Monday, January 4.
University Holiday: Monday, January 18.
Midterm Exam: Friday, February 12.
University Holiday: Monday, February 15.
Last class: Friday, March 12.
Final Exam: Wednesday, March 17, 3-6 PM.

Lecture Summaries

Lecture 1: Outline of the course and motivation. The Weierstrass and Mittag-Leffler problems. Infinite products - PDF
Lecture 2: Turning convergent products into convergent series. Absolutely convergent products. Products of holomorphic functions and their zeros - PDF
Lecture 3: Products of holomorphic functions and their zeros. Logarithmic derivatives of infinite products - PDF
Lecture 4: Factorization of the sine function. The gamma function - PDF
Lecture 5: Properties of the Gamma function. Identities, residues, Gauss' formula, integral representation - PDF
Lecture 6: The Weierstrass problem. Weierstrass elementary factors. Weierstrass factorization. Any meromorphic function in the complex plane is quotient of two entire functions - PDF
Lecture 7: Weierstrass factorization continued. Rephrasing of Weierstrass. Divisors and divisors of meromorphic functions - PDF
Lecture 8: The Weierstrass problem in arbitrary regions - PDF
Lecture 9: Solution to the Mittag-Leffler Problem - PDF
Lecture 10: Examples illustrating the Mittag-Leffler problem - PDF
Lecture 11: Introduction to Montel's theorem. Motivation. Normal families and some examples. Statement of Montel's theorem - PDF
Lecture 12: Equicontinuity and examples. Arzela-Ascoli theorem stated. Obtaining Montel's theorem from Arzela-Ascoli - PDF
Lecture 13: Proof of the Arzela-Ascoli theorem via Cantor diagonalization - PDF
Lecture 14: Statement of Riemann mapping theorem. Some guiding questions - PDF
Lecture 15: Schwarz Lemma and its proof. Automorphisms of the unit disc. Applications to fixed points - PDF
Lecture 16: Applications of Schwarz Lemma. Schwarz-Pick Lemma. Automorphisms of the punctured disc - PDF
Lecture 17: More on automorphism groups. Introduction to Riemann Mapping Theorem - PDF
Lecture 18: Strategy of the proof of Riemann Mapping Theorem. Construction of an injective function which maximizes the derivative - PDF
Lecture 19: Conclusion of the proof of Riemann Mapping Theorem. Remarks about the extremal function. Simply connected domains and taking roots - PDF
Lecture 20: Example of biholomorphism between the half disc and the unit disc. Extension to the boundary. Introduction to Schwarz reflection principle - PDF
Lecture 21: Proof of Schwarz reflection principle. Applications to biholomorphisms of rectangles - PDF
Lecture 22: Introduction to Runge's theorem for compact sets. Polynomial and rational approximation - PDF
Lecture 23: Runge's theorem for compact sets. Strategy for the proof. Cauchy's integral formula for compact sets - PDF
Lecture 24: Finishing the proof of Runge's theorem. Approximation by rational functions from Cauchy's integral formula. Pushing poles to the prescribed location - PDF
Lecture 25: Polynomial approximation and converse to Little Runge. Runge's theorem for open sets - PDF
Lecture 26: Several topological and analytic characterizations of simply connected regions - PDF

All lecture notes can be found here.

Homework:

Homework 1 due Friday, January 15 - PDF. Solutions PDF.
Homework 2 due Friday, January 22 - PDF. Solutions PDF.
Homework 3 due Friday, January 29 - PDF. Solutions PDF.
Homework 4 due Friday, February 5 - PDF. Solutions PDF.
Homework 5 due Friday, February 26 - PDF. Solutions PDF.
Homework 6 due Friday, March 5 - PDF. Solutions PDF.
Homework 7 due Friday, March 12 - PDF. Solutions PDF.

Exams:

Midterm - PDF. Solutions PDF.
Final Exam Topics and Practice - PDF.
Office Hours for the Final: Tuesday 2-4 (Dragos), Tuesday 4-6 (Shubham).
Final - PDF. Solutions PDF.