Time: Tuesdays, Thursdays at 2:15pm.
Rubinstein. Office hours: Tuesday 3:30-5:30pm, Thursday 12-1pm in 382-F.
CA: J. Perea. Email: jperea "at" math.stanford.edu Office hours: Tuesday, Thursday 4-6pm in 381-D.
Room: Building 380, room 381-T.
This is the first course (of three) in the 215 sequence "Complex Analysis,
Geometry, and Topology." It is a first-year graduate level course on
The course will be divided roughly into three parts. The first part will
be a quick review of some essential facts from a basic undergraduate
complex analysis course. In the second part we will concentrate on
conformal mappings and give a proof of the Riemann Mapping Theorem. The
third part will include a collection of topics, largely depending on time
constraints, among which we hope to touch upon the following: the Dirichlet problem
for Laplace's equation, univalent functions and Loewner evolution,
Riemann surfaces and the Uniformization Theorem.
T.W. Gamelin, Complex Analysis.
L. Ahlfors, Complex Analysis (3rd Ed.).
Eight homeworks on a weekly basis. Homework from the previous week due on
the next Thursday in-class or by 3:30pm in the CA's mailbox. Homework
solutions by the CA will be posted on this webpage.
There will be one in-class midterm (October 15 in class)
and one take-home final (handed out on Tuesday, December 1 in class,
due Wednesday, December 9, noon, by email to me or in the CA's
the best seven homeworks will each contribute ten percent as will the midterm,
the final will contribute twenty percent.
Review of undergraduate complex analysis I: complex numbers
and the Fundamental Theorem of Algebra, analytic
functions, Cauchy-Riemann equations, conformal maps, harmonic functions
and their basic properties, Cauchy's theorem and Green's theorem, Cauchy's
integral formula, Cauchy estimates, Liouville's theorem.
Review of undergraduate complex analysis II: Proof of the Fundamental
Theorem of Algebra, Morera's Theorem, Goursat's Theorem, reformulation of
Green's Theorem and dbar notation, Pomepeiu's Formula, power series and
radii of convergence, analytic functions and power series, analyticity at
10/1: further correction posted to Prob. 2.
Review of undergraduate complex analysis III: zeros of analytic
functions, Laurent series, isolated singularities: Riemann's theorem
on removable singularities, characterization of poles,
Review of undergraduate complex analysis IV:
Characterization of meromorphic functions
on the Riemann sphere (Chow's theorem for the Riemann sphere), periodic
functions and Fourier series, Residue and Fractional Residue
Theorems, residue calculus, Argument Principle, Rouche's Theorem.
10/7: Prob. 7 has been shortened and Prob. 8 has been modified.
Hurwitz's Theorem, winding numbers, simply-connected domains and their
various characterizations, formulation of the Riemann Mapping Theorem and
a proof of a weak version of it.
Strategy of proof of the Riemann Mapping Theorem, the Schwarz Lemma,
automorphisms of the unit disc, Montel's thesis theorem and the
10/14: some typos corrected in Prob. 10.
October 14, 4:30pm, in: Herrin T 185 (note special time and place)
Pick's version of the Schwarz Lemma, automorphisms of the disc and
some hyperbolic geometry on the unit disc,
conclusion of the proof of the Riemann Mapping Theorem.
Midterm (topics included: Gamelin Ch. I-VIII and topics from HW1-2).
Uniformization of multiply-connected domains.
Uniformization of multiply-connected
Properties of harmonic functions and introduction to the Dirichlet
problem for the Laplace equation on domains in the plane. The Dirichlet
problem on the unit disk and Poisson's kernel.
Subharmonic functions: differential characterization and maximum
The Perron process. Harmonicity of the upper envelope construction.
11/8: Problems 2 and 7b) have been modified.
Barrier functions and regularity of boundary points.
Alternative proof of the Riemann mapping theorem. Completion of the
proof of the uniformization theorem for multiply-connected domains.
Introduction to complex manifolds. The Riemann sphere.
Green's function for domains in the plane.
Green's function for a Riemann surface.
December 2, 5:15pm (note special time and day)
Green's function for a Riemann surface.
Symmetry of Green's function and
Bipolar Green's function.
Uniformization theorem for Riemann surfaces.