Mathematics 633, Spring 2021

Complex Analysis

Instructor: Lenny Ng
Tuesdays, Thursdays 8:30-9:45 am ET, online only
Office hours: TBA and by appointment.

Course information

This course will be held online on Zoom only. I'll post recordings of the Zoom lectures, as well as lecture notes, on Sakai.

We'll be meeting each Tuesday and Thursday from January 21 to April 20, except that there will be no class on February 18 and March 9. The link to our Zoom meetings is on the Sakai site for the course. If you're not registered but want to check out the course, and you're a Duke student, you should be able to join the Sakai site yourself and get the Zoom link that way. Please let me know if you have any issues with this.

Course syllabus

Textbook: Complex Analysis, 3rd edition, by Lars Ahlfors. I am aware that the book is fairly pricey but if cost is an issue, please note that used copies are pretty readily available.

This course will follow the official syllabus for Math 633 as it's been adopted for the qualifying requirement for the Duke math graduate program. Here's that syllabus (without the optional extra topics, which I don't think I'll cover):

Holomorphic (aka complex analytic) functions, including: definition; branch of log; Cauchy-Riemann equations; as conformal maps; Mobius transformations.
Integration, including: power series representations; Cauchy's Estimate; zeros of analytic functions; Liouville's Theorem; Maximum Modulus Theorem; Cauchy's Theorem and Integral Formula; counting zeros; Open Mapping Theorem.
Singularities, including: classification; Schwarz's Lemma and classification of conformal automorphisms of sphere; Laurent Series; Casorati-Weierstrass Theorem; residues; Argument Principle; Rouche's Theorem.
Space of holomorphic functions, including: Hurwitz's Theorem; Montel's Theorem; Riemann Mapping Theorem; infinite products, the gamma function.
Analytic continuation, including: germs and analytic continuation; Monodromy Theorem; Riemann surfaces (examples).

Familiarity with real analysis at the level of Math 532 will be assumed. It is usually not advisable to take Math 633 if you have previously taken Math 333, as there is significant overlap in material (though our course is taught at a more advanced level). Please see the syllabus for more information. If you have questions about prerequisites, please consult me.

There will be weekly homework assignments and a take-home final exam.