(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 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.
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?