Advanced Complex Analysis
Math 213a - Harvard University - Fall 2023
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.
Students are responsible for all topics covered in the readings and lectures.
In-class presence.
All enrolled students are expected to attend lectures regularly,
and participate in discussions.
To maximize engagement, the use of laptops and cell phones during class is not
permitted. Notebooks and tablets may be used for taking notes.
Online resources.
Students are encouraged to use any online resources they choose for study, review and research
related to this course.
However, only materials from this course can be referred to and used for the homework.
E.g. the use of ChatGPT, searching for solutions on the web, etc, is not permitted.
Grades.
Grades will be based on the homework, which includes a `midterm and final' (see below).
The lowest homework grade, other than the midterm and final grade,
will be dropped.
Graduate students who have passed their
quals are excused from a grade for this course.
Homework.
Weekly homework assignments should be submitted through Canvas.
Late homework will not be accepted.
The due date and time are listed on the homework page.
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. They will also be submitted through Canvas.
The final will be due on a date before the end of reading period (Dec. 10, 2023).