Complex Analysis
Math 213a / Tu Th 1011:30 / Science Center 507
Harvard University  Fall 2000
Instructor:
Curtis T McMullen
(ctm@math.harvard.edu)
Course Assistant:
TBA
Required Texts
 Ahlfors,
Conformal Invariants.
McGrawHill, 1973. (Photocopy)
 Nehari,
Conformal Mapping.
Dover, 1975.
 Remmert,
Classical Topics in Complex Function Theory.
SpringerVerlag, 1998.
Supplementary Text
 Ahlfors,
Complex Analysis.
McGrawHill, 3rd Edition.
Prerequisites.
Intended for graduate students.
Prerequesites include differential forms,
topology of covering spaces and a first course in complex analysis.
Topics.
This course will begin with a rapid, rigorous revisitation of
basic complex analysis, followed by more advanced topics.
Possible topics include:

Basic complex analysis

Differential forms, holomorphic functions.

Distributions, the dbar equation.

Cauchy's theorem, power series, residue calculus,
definite integrals, argument principle.

Maximum principle; Hadamard's 3circles theorem

Singularities

Mobius transformations

Rational maps

Automorphism groups of the disk, plane and sphere.

Schwarz lemma

Normal families

Entire and meromorphic functions

Trigonometric functions

Gamma function

Partition function

Zeta function

Weierstrass products

MittagLeffler theorem

Conformal Mappings

Riemann mapping theorem

The area theorem; compactness

SchwarzChristoffel formula

Local connectivity and boundary values

Doublyconnected regions

Bloch's theorem

Elliptic Functions

Weierstrass pfunction

Theta functions

Modular function

Picard theorem

Universal cover of plane regions

Geometric function theory

Capacity

Harmonic measure

Extremal length

Quasiconformal Maps
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.
Grades.
Graduate students who have passed their
quals are excused from a grade for this course.
Grades for other students will be based on homework.
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.
Calendar.
19 Sept (Tu)  First class 
9 Oct (M)  Columbus day  no class 
10 Nov (F)  Veteran's day  no class 
23 Nov (Th)  Thanksgiving  no class 
19 Dec (Tu)  Last class 
2 Jan (Tu)  Reading period begins 
13 Jan (Sat)  Exams begin 
