Complex Analysis, Fall 2015

Course Information

• Information
• Ceiba
• Lectures: Tuesday, 13:20 ~ 15:10 at Astro-Math 305, and Thursday 13:20 ~ 14:10 at Astro-Math 304
• Office hours: Monday 13:20 ~ 14:10 and Tuesday 15:30 ~ 16:20 or by appointment, at Astro-Math 458
• Grading scheme:
  • Homework 30%
    You have three jokers: the lowest three grades will be discarded.
  • Midterm 30%
  • Final Exam 40%
• References:
  • [A] Lars Ahlfors, Complex analysis. An introduction to the theory of analytic functions of one complex variable. Third edition. MR
  • [S] Elias Stein and Rami Shakarchi, Complex Analysis. MR
  • [N] Rolf Nevanlinna and Veikko Paatero, Introduction to complex analysis. Second Edition MR
  • [C] John Conway, Functions of one complex variable. Second edition. MR
  • [G] Theodore Gamelin, Complex analysis. MR

Lecture summaries and references

• (Week 1) basics of analytic functions: Cauchy--Riemann equation, rational function, power series. Reference: [A, §1 and §2 of ch.2], note.
• (Week 2) complex integration: Cauchy--Goursat theorem, Cauchy integral formula. Reference: [A, §1 and §2 of ch.4], note.
• (Week 3) local structure: Taylor's theorem, zeros and poles. (Due to Typhoon Dujuan, there is no class on Tuesday, and we have two hours lecture on Thursday.) Reference: [A, §2.3, §3.1 and §3.2 of ch.4], note.
• (Week 4) local structure: essential singularity, Taylor and Laurent series, open mapping theorem, maximum principle. Reference: [A, §1 of ch.5 and §3.3, 3.4 of ch.4], note.
• (Week 5) residue and argument principle. Reference: [A, §4, 5 of ch.4], note.
• (Week 6) sums, products and Gamma function. Reference: [A, §2 of ch.5], note.
• (Week 7) Gamma function (continued, and see the note of week 6), entire function: Jensen's formula. Reference: [A, §2.5 and §3.1 of ch.5], note.
• (Week 8) entire function: Hadamard factorization theorem. Reference: [A, §3.2 of ch.5], note;
  Riemann zeta function. Reference: [A, §4 of ch.5], note.
• (Week 9) Prime number theorem. Reference: [Lang, Complex analysis. 4th edition, ch.XVI], note.
  Midterm: 1:00 ~ 3:30pm, November 12, solution.
• (Week 10) normal family. Reference: [A, §5 of ch.5], note.
• (Week 11) fall break.
• (Week 12) Riemann mapping theorem, boundary behavior of conformal mapping. Reference: [A, §1 of ch.6], note.
• (Week 13) conformal mapping to polygons, more on harmonic functions. Reference: [A, §2 and §3 of ch.6], note.
• (Week 14) Dirichlet problem, subharmonic function and Perron's method, conformal mapping of annulus-type regions. Reference: [A, §4 and §5 of ch.6], note.
• (Week 15) conformal mapping of multiply-connected regions, application of normal family. Reference: [A, §5 of ch.6], [G, ch. XII], note.
• (Week 16) Picard's theorems, introduction to complex dynamics. Reference: [G, ch. XII], note.
• (Week 17) basic properties of Julia set and Mandelbrot set, introducing Riemann surfaces. Reference: [G, ch. XII], the note of week 16, and the note.
• (Week 18) Final 1:00 ~ 3:30pm, January 14, solution.

Homework

• Homework 01 (due September 22): [p.28, #3], [p.28, #4], [p.32, #1], [p.32, #2], [p.41, #2].
• Homework 02 (due October 1): [p.108, #6], [p.108, #7], [p.109, #8], [p.120, #1], [p.120, #2], [p.120, #3].
• Homework 03 (due October 6): [p.123, #1], [p.123, #2], prove the fundamental theorem of algebra: any non-constant polynomial admits a root, what is the range of exp(1/z) on the complex plane?
• Homework 04 (due October 13): file.
• Homework 05 (due October 20): file.
• Homework 06 (due October 27): file.
• Homework 07 (due November 3): file.
• Homework 08 (due December 1): file.
• Homework 09: file.
• Homework 10 (due December 8): file.
• Homework 11 (due December 15): file.
• Homework 12 (due December 22): file.
• Homework 13 (due December 29): file.
• Homework 14 (due January 5): file.