Mathematics 633, Spring 2021
Instructor: Lenny Ng
Tuesdays, Thursdays 8:30-9:45 am, online only
Office hours: TBA
and by appointment.
This course will be held online on Zoom only. I'll post
recordings of the Zoom lectures, as well as lecture notes, on Sakai.
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
Space of holomorphic functions, including: Hurwitz's Theorem; Montel's
Theorem; Riemann Mapping Theorem; infinite products, the gamma
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.
will be weekly homework assignments and a take-home final exam.