Advanced Complex Analysis


Math 213a / Tu Th 10-11:30 / Science Center 216
Harvard University - Fall 2017

Instructor: Curtis T McMullen (ctm@math.harvard.edu)

Required Texts
  • Ahlfors, Complex Analysis. McGraw-Hill, 3rd Edition.
  • Nehari, Conformal Mapping. Dover, 1975.
Supplementary Texts
  • Remmert, Classical Topics in Complex Function Theory. Springer-Verlag, 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. 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; Cauchy's formulas
    • Distributions, the d-bar equation
    • Hyperbolic, Euclidean and spherical geometry via Lie groups
    • Schwarz lemma and the Poincare' metric
    • Normal families
  • Entire and meromorphic functions
    • Weierstrass products
    • Mittag-Leffler theorem
    • Trigonometric functions
    • The Gamma function
  • Conformal Mappings
    • Riemann mapping theorem
    • Extremal length
    • Local connectivity and boundary values
    • Doubly-connected regions
    • The area theorem; compactness
    • Schwarz-Christoffel formula
    • Bloch's theorem
    • Picard's theorem
    • Universal cover of plane regions
  • Elliptic Functions
    • Weierstrass p-function
    • 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.


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