Advanced Complex Analysis
Math 213a - Harvard University - Fall 2025
Instructor:
Curtis T McMullen
(ctm@math.harvard.edu)
Required Texts
- Ahlfors, Complex Analysis. McGraw-Hill, 3rd Edition.
- Nehari,
Conformal Mapping.
Dover, 1975.
- Course Notes.
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.Prerequisites 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.
Useful background references: C.H. Edwards, Advanced Calculus of Several Variables, for differential forms; E. L. Lima, Fundamental Groups and Covering Spaces .
Description.
A second course on complex analysis: analytic and meromorphic
functions on the unit disk, the plane, the 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.
Electronics.
To maximize engagement, the use of laptops and cell phones during class is not
permitted. Notebooks and tablets may be used for taking notes.
Homework.
There will be weekly problem sets, to be submitted by Canvas.
Legible handwritten solutions and diagrams are strongly encouraged.
Late homework will not be accepted --- but the lowest homework score will be dropped.
Collaboration between students is encouraged, but you must write your own solutions, understand them and give credit to your collaborators.
Please rely exclusively on course materials (books, notes and lectures).
The use of algebra systems such as Mathematica is allowed, but must be noted on homework.
Grades.
Graduate students who have passed their
quals are excused from a grade for this course.
For others, grades will be based on homework.
There will be two `free solo' homeworks, to be completed without collaboration,
and to take the place of a midterm and a final.
The final homework will be assigned by Tuesday, 2 December and due by midnight Tuesday, 9 December.
AI and online resources.
Only materials from this course can be referred to and used for the homework. E.g. the use of AI, searching for solutions on the web, etc, is not permitted. The use of the many available NI resources is encouraged.