C O M P L E X    A N A L Y S I S ,    F A L L   2 0 1 5

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] Jonh 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.

Up to the homepage of Chung-Jun Tsai