M381D / CSE385S Spring 2014

Complex Analysis, unique # 57085 / # 65915


Instructor: Thomas Chen
Office: RLM 12.138.
Office hours: Fridays, 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 11.176.


Teaching Assistant: Kenny Taliaferro
Office: RLM 11.142.

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 Canvas.

For important dates, see the academic calendar.

Recommended texts:

L.V. Ahlfors, Complex Analysis, McGraw-Hill.
J. B. Conway, Functions of One Complex Variable, Vols I, II. Springer.
R. Remmert, Theory of Complex Functions, Vols. I, II. 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. Stereographic projection.
  2. Contour integration, Cauchy's theorem, Liouville's theorem, Morera's theorem. Harmonic functions, mean value theorem, maximum principle.
  3. 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. Poisson formula, Poisson kernel. Harnack inequality. Approximate identities. Dirichlet problem on the unit disc with continuous and L1 boundary data.
  7. Riemann surfaces, basic definitions and examples. Analytic functions between Riemann surfaces. Valency, degree, genus, Riemann-Hurwitz formula. Elliptic functions. Weierstrass function. SL(2,Z) and fundamental domains.

Homework

  1. HW 1: Due Friday, Jan 24, at the beginning of class.
  2. HW 2: Due Friday, Jan 31, at the beginning of class.
  3. HW 3: Update: Due Monday, Feb 10, at the beginning of class, without Problem 6. Please hand in Problem 6 together with HW 4.
  4. HW 4: Due Monday, Feb 17, at the beginning of class.
  5. HW 5: Due Monday, March 3, at the beginning of class (note that the date has been shifted by 1 week).
  6. Practice set to be discussed in class on Monday, Feb 24.
  7. HW 6: Due Monday, March 17, at the beginning of class.
  8. Additional practice problems.
  9. HW 7: Due Monday, March 24, at the beginning of class.
  10. HW 8: Due Monday, March 31, at the beginning of class.
  11. HW 9: Due Monday, April 14, at the beginning of class.
  12. Practice set to be discussed in class on Monday, April 7.
  13. HW 10: Due Monday, April 21, at the beginning of class.
  14. HW 11: Due Friday, May 2, at the beginning of class.
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 26.

Midterm II on Wednesday, April 9.

Final Exam on Tuesday, May 13. Detailed information on Canvas, please check until May 12 !

The course grade will be determined as follows:

Homework: 20 percent
Midterm (the better out of the two): 30 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.