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Complex Analysis, Fall 2023

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Course Information

• NTU COOL
• Lectures: Tuesday, 13:20 ~ 15:10 and Thursday 13:20 ~ 14:10 at Astro-Math 304
• Office hours: Monday, 14:00 ~ 15:00
• Grading scheme:
  • Homework 30%
    You have two jokers: the lowest two grades will be discarded.
  • Midterm 35%
  • Final Exam 35%
• Teaching Assistant: 連焌凱
• Textbook:
  • [G] Theodore Gamelin, Complex analysis. MR
• Other references:
  • [A] Lars Ahlfors, Complex analysis. An introduction to the theory of analytic functions of one complex variable. Third edition. MR

Lecture summaries and references

• (week 1)
  • complex plane, elementary analytic functions. [G] §I.
  • complex derivative, Cauchy--Riemann equations. [G] §II.2,3 and §II.5,6.
  • fractional linear transformations. [G] §II.7.
  • rational functions. [A] ch.2 §1.4.
• (week 2)
  • line integral, integral theorems on the plane. [G] §III.1,2.
  • harmonic function, harmonic conjugate, mean value theorem. [G] §III.3,4,5.
  • Cauchy--Goursat theorem. [G] §IV.7.
  • Cauchy integral formula, some consequences: derivatives, Morera theorem. [G] §IV.4,6.
• (week 3)
  • Liouville theorem. [G] §IV.5.
  • sequence of functions, power series. [G] §V.2,3,4,5.
  • zeros of an analytic function, open mapping theorem. [G] §V.7.
  • removable of singularity, [G] §VI.2.
• (week 4)
  • Laurent decomposition and Laurent series. [G] §VI.1.
  • classification of isolated singularity. [G] §VI.2.
  • residue theorem. [G] §VII.1.
• (week 5)
  • residue calculus. [G] §VII.2~5.
  • argument principle, Rouche theorem, Hurwitz theorem. [G] §VIII.1~3.
• (week 6)
  • open mapping (revisted) and inverse function. [G] §VIII.4.
  • Schwarz lemma, conformal mapping of the unit disk. [G] §IX.1,2.
  • Poisson kernel of the unit disk. [G] §X.1.
• (week 7)
  • Schwarz reflection principle. [G] §X.2,3.
  • basic conformal maps. [G] §XI.1,2.
  • boundary behavior of conformal maps. [A] ch.6 §1.2.
  • behavior of a conformal map near a vertex of a polygon. [G] §XI.3.
• (week 8)
  • Schwarz--Christoffel formula. [G] §XI.3.
  • Proof of the Riemann mapping theorem. [G] §XI.5,6.
  • Midterm
• (week 9)
  • more examples of conformal maps. [G] §XI.1.
  • normal family of holomorphic functions. [G] §XI.5. [A] ch.5 §5.
• (week 10)
  • Marty's Theorem, Zalcman's Lemma. [G] §XII.1.
  • Theorems of Montel and Picard. [G] §XII.2.
  • simple version of Runge's Theorem. [G] §XIII.1.
• (week 11)
  • Runge's Theorem and an application to radial cluster set. [G] §XIII.1.
  • Mittag-Leffler Theorem and examples. [G] §XIII.2.
• (week 12)
  • elliptic functions. [A] ch.7.
• (week 13)
  • infinite product, Weierstrass product theorem. [G] §XIII.3,4.
  • Gamma function. [G] §XIV.1.
• (week 14)
  • Riemann zeta function. [G] §XIV.3.
  • prime number theorem. [G] §XIV.5.
• (week 15)
  • prime number theorem (continued). [G] §XIV.5.
  • Hadamard theorem, Jensen's formula. [A] ch.5 §3.

Homework

• Homework 1: due September 12
  [p.4 #1 (i) and (j)], [p.10 #2 (b) and (c)], [p.24 #3], [p.31 #2], [p.46 #3], [p.46 #4], [p.46 #5], [p.50 #2], [p.50 #4], [p.57 #1 (d)].
• Homework 2: due September 19
  [p.68 #4], [p.69 #12], [p.82 #2], [p.82 #6], [p.84 #1 (b)], [p.84 #3], [p.84 #4], [P.106 #2 (b)], [p.106 #3 (b)], [p.109 #4].
• Homework 3: due September 26
  [p.111 #1], [p.119 #2], [p.119 #4], [p.122 #2], [p.143 #2 (d)], [p.143 #4], [p.148 #10].
• Homework 4: due October 3
  [p.151 #4], [p.151 #5], [p.157 #1 (e)], [p.157 #3], [p.170 #1 (c)], [p.176 #1 (i)], [p.176 #3].
• Homework 5: due October 11
  [p.198 #4], [p.199 #6], [p.203 #9], [p.205 #7], [p.208 #3], [p.212 #6], [p.215 #7].
• Homework 6: due October 17
  [p.228 #2], [p.230 #2], [p.230 #3], [p.235 #4].
• Homework 7: due October 31
  [p.262 #5], [p.265 #1], [p.265 #9], [p.266 #10], [p.287 #8], and one more on NTU COOL.
• Homework 8: due November 7
  [p.293 #3], [p.293 #4], [p.294 #8], [p.294 #10], [p.302 #5].
• Homework 9: due November 14
  [p.309 #1], [p.309 #3], [p.310 #6], [p.310 #7], [p.319 #3], [p.319 #4], [p.319 #9].
• Homework 10: due November 21
  [p.323 #4], [p.323 #5], [p.346 #3], [p.346 #5], [p.346 #6], and one more on NTU COOL.
• Homework 11: due November 28
  [p.351 #1], [p.351 #4], [p.351 #6], [p.352 #7], [p.352 #11], and [A: p.274 #1].
• Homework 12: due December 5
  [p.356 #10], [p.357 #12], [p.357 #15], [p.360 #1], [p.360 #5], [p.360 #6].
• Homework 13: due December 12
  [p.364 #1], [p.364 #2], [p.364 #5], [p.375 #1], [p.375 #2], [p.375 #4].