Complex analysis
Math 520, USC Fall 2025

Instructors.

• Kyler Siegel (kyler.siegel@usc.edu)

Teaching assistants.

• Yasin Uskuplu (uskuplu@usc.edu)

Time and location.

11am - 11:50am on Mondays, Wednesdays, and Fridays in 414 Kaprielian Hall (KAP). Note: the first class will exceptionally meet in KAP 265.

Office hours.

• Kyler: Kyler: typically 1:30-2:30pm in KAP 400C (these may get modified by an announcement from time to time)
• Yasin Uskuplu: Wednesdays 5-7pm over Zoom (meeting id 983 4250 5194)

About.

This course is an introduction to graduate level complex analysis. The main objects of study are functions of one complex variable which are holomorphic, i.e. complex differentiable. At first glance one might expect the whole theory to closely resemble ordinary calculus of a single real variable, but in fact holomorphic functions have many remarkable features not present in the real case. In the first part of the course we will develop many fundamental ideas in complex analysis, such as holomorphic and meromorphic functions, Cauchy's theorem, integration via residues, entire functions, the Fourier transform, the gamma and zeta functions, and so on. We will then move on to more advanced topics such as the prime number theorem, conformal mappings, elliptic functions, Riemann surfaces, and more. The topics we get to in the latter part of the course will depend partly on the pace and interests of the audience.

Prerequisites.

We will assume familiarity with real analysis. Some knowledge of topology will also be helpful.

Topics.

• complex numbers, the complex plane, and point set topology
• holomorphic functions, power series
• integrals along curves and Cauchy's theorem
• mermorphic functions, residues, and the singularity trichotomy
• definite integrals and contour integration, the argument principle
• the Fourier transform
• entire functions, infinite product representations
• the gamma and zeta functions

• the prime number theorem
• conformal mappings
• elliptic functions
• theta functions and sums of squares
• Riemann surfaces.

Textbooks and references.

• Complex analysis - Stein and Shakarchi [SS]

Problem sets.

There will be weekly problem sets. These will be listed on this website (see the schedule below) and will be handed in via Gradescope. Late homeworks will not be accepted without prior approval from the instructor.

Much of the course material will be developed in the problem sets. It is very important to do all of the problem sets to the best of your ability and to challenge yourself to solve the problems on your own, as this is the most effective way to absorb the material. We expect students to devote a significant amount of time to the problem sets.

While working on the problem sets, you are allowed to collaborate with your peers and to consult textbooks or other references. However, you must write down attributions for any peer, textbook, website etc from which you took any significant ideas. Moreover, you must attempt all problems on your own and your submitted solutions must be written out originally and individually. Submissions which are copied or suspiciously similar are subject to being rejected and potential disciplinary action.

Final presentations.

Grading scheme.

Tentative schedule