Complex Analysis

Math 213a / MWF 12-10 / Science Center 216
Harvard University - Fall 2006
Instructor: Curtis T McMullen (ctm@math.harvard.edu)

Required Texts
  • Ahlfors, Complex Analysis. McGraw-Hill, 3rd Edition.
  • Nehari, Conformal Mapping. Dover, 1975.
  • Remmert, Classical Topics in Complex Function Theory. Springer-Verlag, 1998.
Supplementary Texts
  • Needham, Visual Complex Analysis. Oxford University Press, 1997.
  • Serre, A Course in Arithmetic. Springer-Verlag, 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
    • Distributions, the d-bar equation
    • Schwarz lemma and hyperbolic geometry
    • 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.
18 Sept (M) First class
9 Oct (M) Columbus day -- no class
10 Nov (F) Veteran's day -- no class
24 Nov (F) Thanksgiving -- no class
18 Dec (M) Last class
2 Jan (Tu) Reading period begins
12 Jan (F) Exams begin

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