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" 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: Th 4:00 - 6:00

Online teaching:

We will use Zoom for both lectures and virtual office hours. 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 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.

Grading:

There will be a Midterm on Friday, November 6, as well as a Final Exam on Friday, December 18, 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 at 5pm on Gradescope. In order not to interfere with lecture time, you may submit the homework until Saturdays at 5pm. The grades are recorded in Canvas. Please make sure that your grades are properly recorded.

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

Important dates:

First class: Friday, October 2.
Midterm Exam: Friday, November 6.
Veterans Day: Wednesday, November 11.
Thanksgiving: Thursday-Friday, November 26-27.
Last class: Friday, December 11.
Final Exam: Friday, December 18, 3-6 PM.

Lecture Summaries

Lecture 1: Complex differentiable functions. Comparison with real differentiability. Cauchy-Riemann equations. Harmonic functions - Notes
Lecture 2: Geometric interpretation of differentiability. Conformal maps. Power series. Radius of convergence. Analytic functions. Differentiability of power series. Logarithm - Notes
Lecture 3: More on power series and logarithm. Principal branch, branch cuts. Mobius transformations. Cayley transforms. Biholomorphisms. Generalized circles - Notes
Lecture 4: Mobius transformations and generalized circles. Triple transitive action on the extended plane. Integration over continuously differentiable paths. Basic properties and examples - Notes
Lecture 5: Existence of primitives over arbitrary regions and over discs. Goursat's integral lemma - Notes
Lecture 6: Improvements of Goursat's theorem. Local form of Cauchy's integral formula. Winding numbers - Notes
Lecture 7: Winding number. Homotopies. Invariance of the integral of holomorphic functions over homotopic paths (Cauchy's theorem). Existence of primitives and branches of the logarithm in simply connected sets - Notes
Lecture 8: Homotopy invariance and Cauchy's theorem. Cauchy's integral formula - Notes
Lecture 9: Taylor expansion of holomorphic functions. Cauchy's integral formula for derivatives. Cauchy's estimates - Notes
Lecture 10: Liouville's theorem. Fundamental theorem of algebras. Zeros of holomorphic functions. Order of zeros. The identity theorem - Notes
Lecture 11: Open Mapping Theorem and examples - Notes
Lecture 12: Maximum modulus principle. Laurent series. Holomorphic functions on annuli and Laurent expansion - Notes
Lecture 13: Singularities and their type. Removable singularity theorem. Poles. Meromorphic functions. Essential singularities. Casorati-Weierstrass theorem - Notes
Lecture 14: Methods of computing residues. The residue theorem. Toy example and additional remarks - Notes
Lecture 15: Chains. Nullhomogous chains and connection with nullhomotpic chains. Enhancemments of Cauchy's theorem and CIF. The proof of the residue theorem - Notes
Lecture 16: Proof of CIF for nullhomologous paths. Applications of the residue theorem. Trigonometric integrals - Notes
Lecture 17: Applications of the residue theorem to real analysis. Integration of rational functions. Fourier integrals. The case of poles on the real line - Notes
Lecture 18: More on Fourier integrals and the case of poles on the real line. Logarithmic integrals. Introduction to Mellin transforms - Notes
Lecture 19: More on Mellin transforms. Singularities and residues at infinity - Notes
Lecture 20: More on residues at infiniy. Residue theorem on the extended complex plane. Towards the argument principle - Notes
Lecture 21: The argument principle and generalizations. Applications of the argument principles to inverse functions. Elliptic functions - Notes
Lecture 22: Applications of the argument principle: zeroes and poles of elliptic functions. Rouche's theorem. Idea, statement and examples - Notes
Lecture 23: Proof of Rouche's theorem and more applications. Types of convergence for holomorphic functions. Local uniform convergence and uniform convergence on compact sets - Notes
Lecture 24: Proof of Weierstrass' theorem. Series of holomorphic functions. Hurwitz's theorem and consequences - Notes

All lecture notes can be found here.

Homework:

Homework 0 due Monday, October 5 - PDF.
Homework 1 due Friday, October 9 - PDF. Solutions PDF.
Homework 2 due Friday, October 16 - PDF. Solutions PDF.
Homework 3 due Saturday, October 24 - PDF. Solutions PDF.
Homework 4 due Saturday, October 31 - PDF. Solutions PDF.
Homework 5 due Saturday, November 14 - PDF. Solutions PDF.
Homework 6 due Wednesday, November 25 - PDF. Solutions PDF.
Homework 7 due Friday, December 11 - PDF. Solutions PDF.

Preparation for Midterm and Final:

Practice Problems for Midterm - PDF.
Midterm from Fall 2016 (adapted) - PDF. Solutions PDF.
Midterm - PDF. Solutions PDF.
Final from Fall 2016 (adapted) - 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 - PDF. Solutions PDF.