Complex Analysis II

Spring Semester 2017



Instructor: Matt Kerr
Office: Cupples I, Room 114
e-mail: matkerr [at] math.wustl.edu
Office Hours: Tu/Th 10-11, and by appointment

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. This semester we will cover the remainder 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 then 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 Tuesday and Thursday from 2:30-4 in Duncker Hall Room 101. First class is Tuesday Jan. 17 and last class is Thursday April 27, with two days off for spring break (Tues/Thurs March 14/16).

Midterm (take-home) Exam: distributed April 4 in class; collected April 6 in class
Final (Qualifying, 3 hour) Exam: Tuesday, May 9, 2:30-5:30, in Cupples I Rm. 199

The midterm will cover approximately the first two units above. The qualifying exam will cover material from Math 5021 and Math 5022, and everyone must take it at the time/date listed above.

Assignments:

Posted here each Thursday, collected 1 week later (at beginning of class on Thursday), and returned the following Tuesday. (If you prefer, you may also turn it in to Xiaoyu Dai's mailbox in the department mailroom by 4 PM on Friday.) Solutions will also be posted and will include students' work. Please feel free to come to office hours to discuss problem sets. Make sure you can do the problems I don't assign.

Problem Set 1 / (solutions)
Problem Set 2 / (solutions)
Problem Set 3 / (solutions)
Problem Set 4 / (solutions)
Problem Set 5 / (solutions)
Problem Set 6 / (solutions)
Problem Set 7 / (solutions)
Problem Set 8 / (solutions)
Problem Set 9 / (solutions)
Problem Set 10 / (solutions)
Problem Set 11: not to hand in
PROBLEM SESSION PROBLEMS

Grader: Xiaoyu Dai
Office: Cupples I, Room 213
e-mail: xiaoyu.dai [at] wustl.edu

Lecture Notes:

Will be scanned and posted here as I write them. The hope is that this makes taking notes optional. Here for your convenience is a link to the notes from last term. After some lectures I'll post "addenda" which are related (expository) journal articles, on history or recent developments. Some of these links may only be accessible on campus; if you can't access one (for free) at all, let me know.

Lecture 1: Normal families
Lecture 2: Riemann mapping theorem
Lecture 3: Extension to the boundary
Lecture 4: Explicit conformal mappings (addendum)
Lecture 5: Harmonic functions revisited (addendum)
Lecture 6: Subharmonic functions
Lecture 7: The Dirichlet problem
Lecture 8: Multiply connected regions
Lecture 9: The Gamma function
Lecture 10: Gamma and zeta
Lecture 11: More on zeta functions
Lecture 12: The prime number theorem
Lecture 13: An application of the PNT (addendum)
Lecture 14: Elliptic functions I
Lecture 15: Elliptic functions II
Lecture 16: Addition theorems and modular forms
Lecture 17: More on modular forms
Lecture 18: The Picard theorems
Lecture 19: The Bloch and Landau constants
Lecture 20: Schlict functions I
Lecture 21: Schlict functions II (addendum)
Lecture 22: Green's functions on Riemann surfaces
Lecture 23: The Uniformization Theorem
Lecture 24: Several complex variables
Lecture 25: Poincare and Hartogs
Lecture 26: The Bergman kernel

Books:

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

is the recommended textbook, which means I will follow it at least half the time and some of the problems I assign will come from it. 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. Note that the qualifying exam is three hours, and only part of it (approximately 2/3) is the final exam for this class. (The other part relates to 5021.)

Homework and examination grades will be regularly updated on blackboard.