Complex Analysis

Math 213a / Tu Th 10-11:30 / Science Center 507
Harvard University - Fall 2000
Instructor: Curtis T McMullen (ctm@math.harvard.edu)

Course Assistant: TBA

Required Texts
  • Ahlfors, Conformal Invariants. McGraw-Hill, 1973. (Photocopy)
  • Nehari, Conformal Mapping. Dover, 1975.
  • Remmert, Classical Topics in Complex Function Theory. Springer-Verlag, 1998.
Supplementary Text
  • Ahlfors, Complex Analysis. McGraw-Hill, 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 d-bar equation.
    • Cauchy's theorem, power series, residue calculus, definite integrals, argument principle.
    • Maximum principle; Hadamard's 3-circles 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
    • Mittag-Leffler theorem
  • Conformal Mappings
    • Riemann mapping theorem
    • The area theorem; compactness
    • Schwarz-Christoffel formula
    • Local connectivity and boundary values
    • Doubly-connected regions
    • Bloch's theorem
  • Elliptic Functions
    • Weierstrass p-function
    • 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

Course home page: http://math.harvard.edu/~ctm/math213a