Math 185

Math 185: Complex Analysis, Fall 2015

Instructor: Nikhil Srivastava, email: firstname at math.berkeley.edu

Lectures: MWF 4-5pm, 9 Evans Hall.

Office Hours: Tuesday and Wednesday, 10:15am-11:45am, 1035 Evans Hall.

GSI: Christopher Wong, office hours MW 10:30am-12:30pm and TTh 4:30pm-6:30pm, 1039 Evans Hall.

Text: Brown and Churchill, Complex Variables and Applications, 9e.

Supplementary Text: Gamelin, Complex Analysis. Available online for Berkeley students at Springerlink.

Syllabus: We will cover the first 9 chapters of Brown and Churchill and some additional topics, such as infinite products and applications of conformal mapping.

Piazza: This is an excellent place to ask and answer questions about the material. The website is here and you can sign up here.

Announcements

(8/26) Welcome to the class! The midterms will be on 9/23 and 10/30.
(9/19) I will have extra office hours on Monday, 9/21, from 10:15-11:45.
(9/19) Here is a sample Midterm 1.
(9/22) Here are the sample midterm 1 solutions + handwritten calculation.
(10/5) Midterm 1 and solutions.
(10/19) There will be no office hours on 10/20. Instead, I will have office hours on 10/22 from 1015-1145.
(10/19) You are allowed and encouraged to bring one page (one-sided) of notes for Midterm 2.
(10/26) Here is a sample Midterm 2.
(10/29) Here are the sample Midterm 2 solutions.
(11/09) Here is Midterm 2 and solutions.
(12/12) Here is a Sample Final Exam and solutions.
(12/18) Final Exam and Solutions.

Readings and Homework Schedule

#	Date	Topic	Readings	HW	Notes
1	W 8/26	introduction, complex numbers	1-8
2	F 8/28	Euler's identity, arguments, roots. regions, functions, stereographic projection.	9-14	HW1 out
3	M 8/31	limits, continuity, differentiation.	15-20
4	W 9/2	Cauchy-Riemann equations, chain rule	21-24, lecture notes on asymptotic notation and the chain rule
5	F 9/4	an application to infinite series.	Prof. Hammond's notes	HW1 due HW2 out	Guest lecture by Prof. Alan Hammond
	M 9/7	*holiday, no class*
6	W 9/9	analyticity, Laplace's equation and harmonic functions.	25-27	HW1 solutions
7	F 9/11	exp, log, powers, trig functions	30-40	HW2 due HW3 out
8	M 9/14	integration	41-47
9	W 9/16	antiderivatives	48-49	HW2 solutions
10	F 9/18	Cauchy-Goursat I	50-53	HW4 out
11	M 9/21	Cauchy-Goursat II, review	50-53	HW3 solutions
12	W 9/23	Midterm 1
13	F 9/25	Cauchy Goursat III, deformation theorem	50-53	HW4 due
14	M 9/28	Cauchy integral formula	54	HW5 out.
15	W 9/30	extension of CIF, Morera's theorem, Liouville's Theorem	55-58	rewrite due HW4 solutions
16	F 10/2	fundamental theorem of algebra, max modulus principle	58-59	HW5 due HW6 out
17	M 10/5	sequences, series	60-63	HW5 solutions
18	W 10/7	Taylor's theorem, uniform convergence	62-63, 69-70
19	F 10/9	Manipulation of series, examples	71-73, 64	HW6 due HW7 out
20	M 10/12	zeros of analytic functions, identity principle, Laurent series	82, 28, 65-68
21	W 10/14	Proof of Laurent expansion	65-68
22	F 10/16	isolated singularities	74, 78, 83-84	HW8 out HW6 solutions
23	M 10/19	residue theorem, residues at poles	75-81
24	W 10/21	argument principle	93
25	F 10/23	Rouche's theorem, open mapping theorem	94 lecture notes	HW9 out HW7 solutions
26	M 10/26	applications to definite integrals	92, 85-87	HW8 solutions
27	W 10/28	review for midterm	50-84, 93-94	HW9 solutions +handwritten
28	F 10/30	Midterm 2
29	M 11/2	Jordan's lemma, indented contours	88-89	HW10 out
30	W 11/4	integration around a branch point / through a branch cut	90-91		Guest lecture by Prof. Gang Liu
31	F 11/6	the Basel problem, the point at infinity	lecture notes
32	M 11/9	residues at infinity, the Fibonacci sequence	lecture notes	HW11 out
	W 11/11	*holiday*		HW10 solutions
33	F 11/13	Mobius Transformations	96-101	HW12 out watch this video
34	M 11/16	Conformal Mapping	102-103, 112-113
35	W 11/18	Inverse Function Theorem, Harmonic Functions	114,115	HW11 solutions
36	F 11/20	Dirichlet Problem	116-117, 119 supplementary notes	HW13 out
37	M 11/23	Analytic Continuation, Gamma Function	lecture notes
38	M 11/30	Prime Number Theorem	lecture notes	HW14 out
39	W 12/2	Prime Number Theorem		HW 13 due HW12 solutions
40	F 12/2	Prime Number Theorem / review	interesting article	HW13 solutions

Please read the assigned sections before class. All section numbers are from Brown and Churchill, except those marked with a G, which are from Gamelin.

Grading: 20% Homework, 40% Midterms, 40% Final. The lower midterm score will be replaced by the final exam score, if it helps.

Homework will be assigned every Friday and due the following Friday at the end of class. Solutions to selected problems will be posted on this page, and late homework will not be accepted.

Michael Hutchings on writing proofs (which will be required) and on academic dishonesty (which will not be tolerated).
Writing homework solutions in LaTeX is encouraged.