M381D / CSE385S Spring 2013

Complex Analysis, unique # 56850 / # 65305


Instructor: Thomas Chen
Office: RLM 12.138.
Office hours: Mondays, 12:00 - 12:50 PM.
Email: t c A_T m a t h . u t e x a s . e d u
Lectures: MWF, 10:00 - 10:50 AM
Location: RLM 9.166.


Teaching Assistant: Kenny Taliaferro
Office: RLM 11.142.
Office hours: Tuesdays, 11:00 - 11:50 AM.


Syllabus and Course Information


This is a graduate course on Complex Analysis. We will cover the material listed on the Preliminary Exam Syllabus in Complex Analysis, and some additional topics.

Prerequisites: Familiarity with the subject matter of the undergraduate analysis course M365C, a syllabus of which can be found at the end of the page linked here.

Updated course information will be posted here and on Blackboard.

For important dates, see the academic calendar.

Recommended texts:

L.V. Ahlfors, Complex Analysis.
R. Remmert, Theory of Complex Functions, Springer.
E. M. Stein, R. Shakarchi, Complex Analysis, Princeton Lectures in Analysis.
W. Rudin, Real & Complex Analysis, McGraw-Hill.

Online resources:

C. McMullen, online course notes on complex analysis.
C. Teleman, online course notes on Riemann surfaces.


Schedule:
  1. Complex differentiation, Cauchy-Riemann equations, conformality. Holomorphic functions, analytic functions.
  2. Contour integration, Cauchy's theorem, Liouville's theorem, Morera's theorem. Harmonic functions, mean value theorem, maximum principle.
  3. Stereographic projection, Moebius transforms, Schwarz lemma. Automorphisms of the unit disc and of the upper half plane. Holomorphic functions on the Riemann sphere.
  4. Isolated singularities and residues, meromorphic functions, Laurent series. Winding number, cycles, null homology, basics on differential forms, generalized Cauchy theorem, residue theorem. Uniqueness theorem, analytic continuation.
  5. Convergence and normal families. Mittag-Leffler theorem, Weierstrass and Hadamard factorization theorems, order and genus of entire functions. Riemann mapping theorem.
  6. Riemann surfaces, basic definitions and examples. Analytic functions between Riemann surfaces. Valency, degree, genus, Riemann-Hurwitz formula. Riemann surfaces as quotients, universal coverings, little Picard theorem.
  7. Elliptic functions. Weierstrass function. Hyperbolic plane and geodesics of the Poincare metric. SL(2,Z) and fundamental domains.
  8. Poisson formula, Poisson kernel. Harnack inequality. Approximate identities. Dirichlet problem on the unit disc with continuous and L1 boundary data.

Homework

  1. HW 1: Due Friday, Jan 25, at the beginning of class.
  2. HW 2: Due Friday, Feb 1, at the beginning of class.
    Last modified on Jan 25, 3:17 PM.
  3. HW 3: Due Friday, Feb 8, at the beginning of class.
    Last modified on Feb 1, 9:35 AM.
  4. HW 4: Due Friday, Feb 15, at the beginning of class.
    Last modified on Feb 10, 2:25 PM.
  5. HW 5: Due Friday, Feb 22, at the beginning of class.
    Last modified on Feb 15, 3:30 PM.
  6. Practice Problems. They will be discussed on Monday, Feb 25, in class.
    Last modified on Feb 25, 3:30 PM.
  7. Midterm 1 on Wednesday, Feb 27, in the usual classroom, at the usual time.
    Exam info will be posted on Blackboard.
  8. HW 6: Due Friday, Mar 8, at the beginning of class.
    Solution corrected on Mar 21, 7:15 AM.
  9. HW 7: Due Friday, Mar 22, at the beginning of class.
    Solution corrected on Mar 27, 8:25 AM.
  10. HW 8: Due Friday, Mar 29, at the beginning of class.
    Last modified on Mar 23, 1:05 PM.
  11. HW 9: Due Friday, April 5, at the beginning of class.
  12. Midterm 2 on Wednesday, April 10, in the usual classroom, at the usual time.
    Exam info will be posted on Blackboard.
  13. HW 10: Due Friday, Apr 19, at the beginning of class.
    Last modified on Apr 14, 1:50 AM.
  14. HW 11: Due Friday, Apr 26, at the beginning of class.
    Last modified on Apr 21, 11:10 PM. Hints added.
  15. HW 12: Due Friday, May 3, at the beginning of class.
    Last modified on Apr 28, 12:40 PM.
  16. Final Exam on Friday, May 10, 2013. Please see Blackboard for detailed information.
Collaborations are encouraged, but you have to hand in your own solutions. Simply copying somebody else's work is not acceptable.

Exams and Grades


There will be two in-class midterms, and a final exam.

Midterm I on Wednesday, February 27, 2013.

Midterm II on Wednesday, April 10, 2013.

Final Exam on Friday, May 10, 2013. Please see Blackboard for detailed information.

The course grade will be determined as follows:

Homework: 15 percent
Midterm (the better out of the two): 35 percent
Final: 50 percent

Range of letter grades: A, A-, B+, B, B-, C+, C, C-, D+, D, D-, F.

The University of Texas at Austin provides upon request appropriate academic accommodations for qualified students with disabilities.
For more information, contact the Office of the Dean of Students at 471-6259, 471-4641 TTY.