Advanced Complex Analysis
Math 213a / Tu Th 1011:30 / Science Center 216
Harvard University  Fall 2017
Instructor:
Curtis T McMullen
(ctm@math.harvard.edu)
Required Texts
 Ahlfors,
Complex Analysis.
McGrawHill, 3rd Edition.
 Nehari,
Conformal Mapping.
Dover, 1975.
Supplementary Texts
 Remmert,
Classical Topics in Complex Function Theory.
SpringerVerlag, 1998.
 Stein and Shakarchi,
Complex Analysis.
Princeton University Press, 2003.
 Needham,
Visual Complex Analysis.
Oxford University Press, 1997.
 Sansone and Gerretsen,
Lectures on the Theory of Functions of a Complex Variable.
(2 volumes.) P. Noordhoff, Ltd., 1960.
 Serre,
A Course in Arithmetic.
SpringerVerlag, 1973.
 Titchmarsh,
Theory of Functions.
Cambridge, 1939.
Prerequisites.
Intended for graduate students.
Prerequesites include differential forms,
topology of covering spaces and a first course in complex analysis.
Undergraduates require Math 113 and 131, or permission of the instructor.
Description.
A second course on complex analysis on the plane, sphere and complex tori.
Possible topics include:

Basic complex analysis

Holomorphic functions and forms; Cauchy's formulas

Distributions, the dbar equation

Hyperbolic, Euclidean and spherical geometry via Lie groups

Schwarz lemma and the Poincare' metric

Normal families

Entire and meromorphic functions

Weierstrass products

MittagLeffler theorem

Trigonometric functions

The Gamma function

Conformal Mappings

Riemann mapping theorem

Extremal length

Local connectivity and boundary values

Doublyconnected regions

The area theorem; compactness

SchwarzChristoffel formula

Bloch's theorem

Picard's theorem

Universal cover of plane regions

Elliptic Functions

Weierstrass pfunction

Modular function

Theta functions

Partition function

Zeta function
Reading and Lectures.
Students are responsible for all topics covered in
the readings and lectures. Lectures may go beyond the
reading, and not every topic in the reading will be
covered in class.
All enrolled students are expected to attend lectures regularly.
Grades.
Graduate students who have passed their
quals are excused from a grade for this course.
Grades for other students will be based on the homework,
which includes a `midterm and final' (see below).
Homework.
Homework will be assigned every week.
Late homework will not be accepted.
Collaboration between students is encouraged, but you
must write your own solutions, understand them and
give credit to your collaborators.
Midterm and Final.
Two homeworks will be designated the `midterm' and `final'
assignments. These are to be done without collaboration.
