Complex Analysis II

Spring Semester 2023

Instructor: Matt Kerr
Office: Cupples I, Room 114
e-mail: matkerr [at]
Office Hours: 8-9 W (on Zoom), 4-5 F (in office)

Course Outline:

I. Riemann mapping theorem and the Dirichlet problem
II. Elliptic functions and some number theory
III. Additional topics in one complex variable
IV. Introduction to several complex variables

This is the second half of a year-long course which forms the basis for the Ph.D. qualifying examination in complex variables. The material roughly correlates to the second half of Ahlfors (3rd edition), including elliptic functions and the Riemann mapping theorem, as well as the big Picard theorem and prime number theorem. Time permitting, we will also give an introduction to modular forms, followed by several complex variables and the Bergman kernel.

Prerequisites: Math 5021 or permission of instructor.

Class Schedule:

Lectures are on Monday, Wednesday and Friday from 1-1:50 in Lopata 201. First class is Wednesday Jan. 18 and last class is Friday April 87, with three days off for spring break (M/W/F March 13/15/17).

Midterm (take-home) Exam: TBA, possibly take-home
Final (Qualifying) Exam: Wednesday, May 10, 1-3pm, in Lopata 201

The midterm will cover approximately the first two units above. The qualifying exam will cover material from the whole course, and everyone must take it at the time/date listed above.


Posted here by Friday, due the following Thursday by 6PM on Gradescope, and returned by the following Wednesday. Solutions will be posted on Canvas. Please feel free to come to office hours to discuss problem sets.

Problem Set 1 (due Thursday Jan. 26)
Problem Set 2 (due Thursday Feb. 2) [tex]
Problem Set 3 (due Thursday Feb. 9) [tex]
Problem Set 4 (due Thursday Feb. 16) [tex]
Problem Set 5 (due Thursday Feb. 23) [tex]
Problem Set 6 (due Thursday Mar. 2) [tex]
Problem Set 7 (due Thursday Mar. 9) [tex]
Problem Set 8 (due Thursday Mar. 23) [tex]
Problem Set 9 (due Thursday Apr. 6) [tex]
Problem Set 10 (due Thursday Apr. 13) [tex]
Problem Set 11 (due Thursday Apr. 20) [tex]
Problem Set 12 (not to hand in)

Grader: RJ Acuna
Office: Cupples I, Room 213
e-mail: rjacuna [at]

Lecture Notes:

Will be posted here the same day as the lecture. The hope is that this makes notetaking optional.

Lecture 1: Normal families
Lecture 2: Riemann mapping theorem
Lecture 3: Extension to the boundary
Lecture 4: From boundary to interior
Lecture 5: Explicit conformal mappings
Lecture 6: Harmonic functions revisited
Lecture 7: More on harmonic functions
Lecture 8: Subharmonic functions I
Lecture 9: Subharmonic functions II
Lecture 10: The Dirichlet problem
Lecture 11: Applications of Dirichlet
Lecture 12: Multiply connected regions I
Lecture 13: Multiply connected regions II
Lecture 14: Multiply connected regions III
Lecture 15: The Gamma function
Lecture 16: Gamma and zeta
Lecture 17: More on zeta
Lecture 18: Primes and zeta zeroes
Lecture 19: The prime number theorem
Lecture 20: Apery's theorem
Lecture 21: Lattices in C
Lecture 22: Abel's theorem
Lecture 23: Elliptic functions and elliptic curves
Lecture 24: Elliptic addition theorems
Lecture 25: Introduction to modular forms
Lecture 26: More on modular forms
Lecture 27: Modular forms of higher level
Lecture 28: The Picard theorems
Lecture 29: Geometric function theory
Lecture 30: The Bloch and Landau constants
Lecture 31: Schlicht functions I
Lecture 32: Schlicht functions II
Lecture 33: Green's functions on Riemann surfaces I
Lecture 34: Green's functions on Riemann surfaces II
Lecture 35: The Uniformization Theorem
Lecture 36: Several complex variables I
Lecture 37: Several complex variables II
Lecture 38: Several complex variables III
Lecture 39: Several complex variables IV
Lecture 40: The Bergman kernel

Lars Ahlfors, Complex Analysis (3rd Ed.); McGraw-Hill

is the recommended textbook, which means I will follow it some of the time and some of the problems I assign will come from it. I assume most of you have a copy from Math 5021; if not, then buy, check out, or borrow a copy.

If you would like to read more adventurously than Ahlfors and/or my lecture notes, here are some suggestions. First, there are many other excellent standard texts, including

John B. Conway, Functions of One Complex Variable; Springer

Robert Greene and Steven Krantz, Function Theory of One Complex Variable; AMS

and the second half of

Walter Rudin, Real and Complex Analysis (3rd Ed.); McGraw-Hill.

For a point of view based in formal and convergent power series (convenient for locally computing composition inverses and solutions of differential equations) you can consult

Henri Cartan, Elementary Theory of Analytic Functions of One and Several Complex Variables; Addison-Wesley

Serge Lang, Complex Analysis (3rd Ed.); Springer.

For a view toward several complex variables there is

Raghavan Narasimhan and Yves Nievergelt, Complex Analysis in One Variable; Birkhauser ;

and the beautiful expository monograph

Steven Krantz, Complex Analysis: the Geometric Viewpoint; MAA

treats theorems in complex analysis through the prism of differential geometry. Finally,

Harvey Cohn, Conformal Mapping on Riemann Surfaces; Dover

leads (with lots of beautiful pictures and physical intuition) into Riemann surfaces and complex algebraic geometry. I will place one copy of each of these books on reserve at the Olin Library.

Grading Policy:

Your final grade for the semester is determined as follows: HW 40%, midterm 20%, final exam 40%. I will drop the lowest two grades you receive on homework.

Homework and examination grades will be regularly updated on Canvas.

Grades are typically curved in a course like this but will never be less than the following scale:

A+ A A- B+ B B- C+ C C- D F
TBA 90+ [85,90) [80,85) [75,80) [70,75) [65,70) [60,65) [55,60) [50,55) [0,50)

If you are a graduate student, a letter grade of B is required to pass; if you are an undergraduate taking this class Pass/Fail, you must earn a C- to pass.

The Washington University academic integrity policies are here. All work submitted under your name is expected to be your own; please make sure to document any ideas that come from another source.