Course on Complex Analysis Vanderbilt 2018

Syllabus:

Complex integration, calculus of residues, harmonic functions, entire and meromorphic functions, conformal mapping, normal families, analytic continuation,elliptic functions, iteration of rational maps.

Course Description.

I gave this course last year and was very happy with the general structure- a bit more than half the course being the usual topics of an undergraduate course, but from a more sophisticated and rigorous mathematical point of view. Thus for instance we prove Cauchy’s formula and give a treatment of the residue formula over general contours. Topics covered in this part of the course include the maximimum modulus theorem,harmonic functions and harmonic conjugates, Schwarz reflection principle, the argument principle, Taylor and Laurent series, Rouché’s theorem,branches, a little analytic continuation, Schwarz lemma. Then we introduce some new topics including normal families building up to the first major goal of the course, the Riemann mapping theorem that states that any simply connected domain in the complex plane ℂ, different from ℂ is conformally equivalent to the unit disc. The second part of the course begins with a look at elliptic functions based on the Weierstrass ℘ function. If there is time we will give a proof of the Uniformisation theorem. Finally as a special topic we investigate iteration of rational functions where we meet the Julia, Fatou and Mandelbrot sets and learn how to understand pictures like:

trial image

Potential projects

MIDTERM

We will have a midterm (closed book) during the lecture time on Thursday October 4.

Homework

Throughout the course emphasis wll be placed on examples and homework will be assigned regularly. Look for homework assignments HERE

First homework assignment- I was very impressed with the one I assigned last year so here it is again-hand it in on Tuesday 4 September

Second homework assignment, hand in Tuesday September 11

Third homework assignment, hand in Tuesday September 18

Fourth homework assignment, hand in Tuesday September 25

Fifth homework assignment, hand in Tuesday October 2

Sixth homework assignment, hand in Tuesday October 16

Seventh homework assignment, hand in Tuesday October 22

Last homework assignment, hand in after Thanksgiving

My aopologies, for some reason I did not get the above assignment up last week. Just get it done as soon as you can. The last question will play a role in our proof of the Riemann mapping theorem.

Arzela Ascoli1

Arzela Ascoli2

Second elliptic functions notes.

Last year’s course home page:

The home page of the course from last year is here:

Last year’s course home page:

Textbook:

The texts for the course will be the classic “Complex Analysis” by Ahlfors, and Sarason’s “Notes on complex function theory.”

Midterm, grades

So that I have some idea how much students are understanding we will have a midterm in class some time in September.Otherwise grades will be assigned on the basis of a short (2-3 pages) essay on a topic not covered explicity in the course, e.g. “The first three zeros of the Riemann Zeta function.” Topics may be proposed by students or selected from a list that I will give. The topics proposed last year were found to be too open-ended so this year I will give more explicit and detailed topics. The student is supposed to investigate the literature on the topic and the essay will be the result of that investigation. I encourage students to work in groups, and will be happy to discuss progress on the essay with a group.

Office hours

Tentative office hours are Tuesday and Thursday from 3pm to 4pm and may also be arranged by appointment. My office is SC1413

Course on Complex Analysis Vanderbilt 2018