Math 185 : Complex Analysis, Fall 2016

Basic Information

Class: MWF, 3:00-4:00 PM

Instructor: Ved V. Datar

Email: vv lastname at, 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.

Grade Distribution

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.

Special Announcements

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.

Writing Proofs

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.

First midterm and Solutions.

Second midterm and Solutions.

Practice problems for the final and solutions.

Tentative schedule, homeworks and lecture notes

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 theorem37-39 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 45-47Cauchy Integral Formula, Analyticity
19 F 10/07 Morera's theorem, Liouville's theorem, fundamental theorem of algebra 49-50Liouville'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 84-85Meromorphic 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 76-77Residue 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 91Residue Calculus-1
31 F 11/04 definite integrals (cont.) 78-83
32 M 11/07 definte integrals (cont.) 78-83Residue 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 theorem209-212, 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 160-162Special functions
40 F 12/2 Zeta function 168-172

Last modified: Sat Dec 10 00:20:36 PST 2016