Introduction to Complex Analysis
Peter Koroteev
833 Evans, pkoroteev@math.berkeley.edu /~pkoroteev
Lectures: Tues/Thurs 9:30 am - 11:00 am, 70 Evans Hall
Office Hours: Tuesday 11:10 am -- 12:40 pm, Thursday 4:45 pm -- 6:15 pm
Textbooks: main book: `Complex Analysis' by L. V. Ahlfors. Other useful textbook is `Complex Analysis' by S. Lang, `Complex Analysis' by T. Gamelin.
Useful Links: Academic honesty in mathematics courses: (courtesy of Michael Hutchings), How to get an A in this class (courtesy of Kathryn Mann).
Material which will be covered:
Plan of the lectures:
(will be updated throughout the course)
| Week 1 (Jan 22nd and 24th) |
Complex numbers, polar representation, stereographic projection Ahlfors: Chapter 1 |
| Week 2 (Jan 29th and 31st) |
Complex functions, Exp, Log Ahlfors: Section 3 Gamelin: I.4 - I.8 on Riemann surfaces Lang: Sec 5,6 |
| Week 3 (Feb 5th and 7th) |
Holomorphic functions, Harmonic functions. Fractional linear transformations. Lang: Sec 5,6,7; VII. 5 (for FLT) Gamelin: II.1 - II.7 Ahlfors: Section 3 from Chapter 3 |
| Week 4 (Feb 12th and 14th) |
Elementary conformal maps. Fractional linear transformations. Complex Integration. Lang: VII. 5 Ahlfors: Sections 3 and 4 in Chapter 3 |
| Week 5 (Feb 19th and 21st) |
Complex Integration. Cauchy Theorem Ahlfors: 4.1, 4.2 Lang: Chapter III |
| Week 6 (Feb 26th and 28th) |
Review of the material. Midterm 1, no homework will be assigned. |
| Week 7 (March 5th and 7th) |
Winding numbers and Cauchy's theorems. Ahlfors: 4.2, 4.3 Lang: Chapters IV, V |
| Week 8 (March 12th and 14th) |
Taylor, Laurent series, isolated singularities. Complex Integration Ahlfors: Chapter 4, sections 4,5 Lang: Chapters V, VI |
| Week 9 (March 19th and 21st) |
Complex integration, residues, applications of Cauchy formula Lang: Chapters V, VI Ahlfors: Chapter 4, sections 5 |
| Week 10 (April 2nd and 4th) |
Complex integration. Fourier transform, Mellin transform. Lang: Chapter VI |
| Week 11 (April 9th and 11th) |
Review of the material. Midterm 2, no homework will be assigned. Midterm #2 includes various methods of complex integration from weeks 8,9,10 as well as differential forms, see Ahlfors p. 144-146 and Ex 1 on p. 148. Theorems and corollaries in Ahlfors p. 132-137 about maximum principle may become useful as well. |
| Week 12 (April 16th and 18th) |
Entire and Meromorphic functions. Partial fractions, infinite products, Gamma function. Lang: Chapters XIII, XV Sec. 2 Ahlfors: Chapter 5, Section 2 |
| Week 13 (April 23rd and 25th) |
Elliptic Functions Lang: Chapter 14 |
| Week 14 (April 30th and May 1st) |
Optional: Resurgence, Picard Fucks equation https://arxiv.org/pdf/1410.0388.pdf (see sec. 2.3 on p. 15) |
| Week 15 (May 7th and 9th) |
RRR week |
| Final examination May 15th |
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