Class: MWF, 3:00-4:00 PM
Instructor: Ved V. Datar
Email: vv lastname at math.berkeley.edu, no spaces
Office: 1067 Evans Hall
Office hours: MW, 4:00-5:00PM
GSI: Brandon Williams, Evans 732, Office hours - TuTh, 10:00AM-11:59AM and MWF, 1:00PM-3:00PM
Text: Elias Stein and Rami Shakarchi, Complex Analysis, Princeton lectures in analysis
Suplementary reading: Brown and Churchill, Complex variables and applications. For more advanced reading - Complex Analysis by Ahlfors.
Homeworks - 20%, 2 Midterms - 20% each, Final - 40%
There will be 12 homeworks. The best ten will be counted towards the grade. There is no late submission of homeworks.
There will be no make-up exams. The lower of the scores on the midterms will be replaced by the final exam marks (divided by two) or the other mid term score if it helps. This is also applicable in the event of a missed exam (where the missed exam will be counted as 0 points). To pass the class, you have to take the final exam.
If you are a disabled student (with or without a document from the Disabled Students' Program) and require special accommodations of any kind, please e-mail the instructor as soon as possible, and no later than September 10.
If you are representing the university on some official duties (say if you are an athlete or in a band), and if there is a conflict with any of the mid-terms, please let the instructor know before September 10.
To do well in the course, it is important to be able to understand as well as write proofs. It might be useful to read Prof. Hutching's article on mathematical reasoning.
Most of our proofs will use standard set theoretc notation. Here are some notes from Prof. Bergman on set theory.
We will also often use mathematical induction. Here are some notes on that by Prof. Bergman.
Practice problems for first mid-term. Solutions.
Practice problems for second mid-term and solutions.
A useful reference for branch cuts . Also useful might be this video from 29 minutes on to roughly 39 minutes. But beware that the rest of the video talks about Riemann surfaces etc. which is beyond our syllabus. But in those minutes the lecturer talks about branch cuts for (z-a)^{1/2}(z-b)^{1/2}, a special case of which was covered very hurriedly in class today.
Practice problems for the final and solutions.
| Number | Date | Topic | Reading | Homework | Notes |
| 1 | W 8/24 | introduction, review of complex numbers | 1-2 | Introduction, Complex Plane | |
| 2 | F 8/26 | complex plane (cont.), convergence, basic topology | 4-6 | HW-1 (due 9/02). Hints/Solutions | Topology |
| 3 | M 8/29 | limits, continuity, holomorphicity | 8-9 | Limits,Continuity | |
| 4 | W 8/31 | holomorphicity (cont.), power series | 9-10, 14-15 | Holomorphicity | |
| 5 | F 9/02 | power series (cont.) | 16-18 | Power Series | |
| M 9/05 | holiday | HW-2 (due 9/12) Hints/Solutions | |||
| 6 | W 9/07 | exp, trig functions, log, powers | Exp,Trig,Log (edited) | ||
| 7 | F 9/09 | Cauchy Riemann equations | 10-13 | Cauchy-Riemann | |
| 8 | M 9/12 | curves, complex integration | 19-22 | HW-3 (due 9/19) Hints/Solutions | Complex Integration |
| 9 | W 9/14 | primitives | 22-24 | Primitives | |
| 10 | F 9/16 | Cauchy-Goursat theorems | 31-36 | Cauchy-Goursat | |
| 11 | M 9/19 | proof of Goursat's theorem | 37-39 | HW-4 (due 9/26) | |
| 12 | W 9/21 | homotopies, simply connected domains | Homotopy Theorem | ||
| 13 | F 9/23 | simply connected domains (cont.), general Cauchy's theorem | 95-97 | Generalized Cauchy | |
| 14 | M 9/26 | general Cauchy's theorem, applications to definite integrals | 97, 41-45 | HW-5 (due 10/03) | Index |
| 15 | W 9/28 | logorithm revisited, branch cuts | 97-101 | Logarithm | |
| 16 | F 9/30 | branch cuts (cont.), summary | 97-101 | ||
| 17 | M 10/03 | midterm-1 | HW-6 (due 10/10) | ||
| 18 | W 10/05 | Cauchy's integral formula, analyticity | 45-47 | Cauchy Integral Formula, Analyticity | |
| 19 | F 10/07 | Morera's theorem, Liouville's theorem, fundamental theorem of algebra | 49-50 | Liouville's theorem, Fundamental theorem of algebra | |
| 20 | M 10/10 | Sequences of holomorphic functions, zeroes | 47-49, 53 | HW-7 (due 10/17) | Sequences |
| 21 | W 10/12 | zeroes (cont.), analytic continuation | 50-51 | Zeroes of holomorphic functions | |
| 22 | F 10/14 | isolated singularities | 52-55 | Isolated Singularities | |
| 23 | M 10/17 | isolated singularities (cont.) | 71-75 | HW-8 (due 10/24) | |
| 24 | W 10/19 | meromorphic functions, Laurent series | 84-85 | Meromorphic fucntions | |
| 25 | F 10/21 | Laurent series (cont.) | 86-88 | Laurent Series | |
| 26 | M 10/24 | Residue formula | Ex 3 on pg. 109 | HW-9 (due 10/31) | |
| 27 | W 10/26 | Argument principle | 76-77 | Residue theorem and argument principle | |
| 28 | F 10/28 | open mapping, max modulus principle | 89-90 | Applications of argument principle | |
| 29 | M 10/31 | Rouche's theorem | 92 | HW-10 (due 11/07) | |
| 30 | W 11/02 | definite integrals | 91 | Residue Calculus-1 | |
| 31 | F 11/04 | definite integrals (cont.) | 78-83 | ||
| 32 | M 11/07 | definte integrals (cont.) | 78-83 | Residue Calculus-2 (branch cuts) | |
| 33 | W 11/09 | midterm-2 | |||
| F 11/11 | holiday | 96-101 | |||
| 34 | M 11/14 | conformal maps | 206-208 | Conformal Mappings | |
| 35 | W 11/16 | examples, Riemann mapping theorem | 209-212, 224 | HW-11 (due 11/21 or 11/28) | |
| 36 | F 11/18 | Riemann mapping theorem (cont.) | 218 | ||
| 37 | M 11/21 | Riemann mapping theorem (cont.) | 225-227 | ||
| W 11/23 | Thanksgiving | ||||
| F 11/25 | Thanksgiving | ||||
| 38 | M 11/28 | Riemann mapping theorem (cont.) | 228-230 | HW-12 (due 12/05) | Riemann Mapping Theorem |
| 39 | W 11/30 | Analytic continuation, gamma function | 160-162 | Special functions | |
| 40 | F 12/2 | Zeta function | 168-172 |