Riemann sphere, Möbius transformations, Cauchy-Riemann equations, Cauchy's Theorem and Integral Formula, Argument Principle, Residue Theorem, Riemann Mapping Theorem, Schwarz Christoffel Formula
Complex Analysis I	15-Math-604-001	Winter Quarter 2005

This page is a work in progress! (Last revised 30 Sept 2005)

Here are links to pdf files for my notes and a list of problems.

Textbooks There are zillions of books about Complex Analysis available in the Geo-Math-Physics Library. I have put a number of these on open reserve---if you check one out at the end of the day, you can keep it over night. Below is a short list of texts which I think are pretty good. For the most part, I will follow Ahlfors. The books by Carathéodory are very geometric, which I quite like. The book by Greene & Krantz is interesting and takes a novel approach. The books by McGehee and Palka are elementary with lotsa details presented. The books by Greene & Krantz, McGehee, and Palka are all loaded with good problems.

A	Ahlfors	Complex Analysis (third edition)
C	Carathéodory	Theory of Functions (of a complex variable) Vols. I & II (second edition)
GK	Greene & Krantz	Function Theory of One Complex Variable (second edition)
M	McGehee	An Introduction to Complex Analysis
P	Palka	An Introduction to Complex Function Theory

General Syllabus Chapters: 1-4 in A; 1-3,6 in GK; 1-3 in M; I-V,IX in P

Course Goals First and foremost, this is a course geared towards teaching you to think, act, and problem solve like a mathematician. Of particular importance, to me, is your acquiring the ability to communicate mathematical ideas effectively. This means learning how to read, and especially to write, mathematical proofs. We shall accomplish this in the guise of studying elementary complex analysis and the roots of geometric function theory. A subgoal, at least for some of you, is preparation for the preliminary PhD examination in complex analysis; I will have more to say about this in spring quarter (at which time I will run a prelim practice problem session).

Grades Your final course grade will be determined from your performance on a midterm exam, a final exam, your homework scores, and class and problem session participation. Roughly speaking, an A means your work is at the PhD level, a B indicates masters level work, and anything less describes work which is not at the graduate level.

Homework & Problem Session I will have a long list of suggested problems for you to solve. I plan to have a weekly hour long (90 minutes if possible!) problem session. Here I will expect you, the students, to present solutions to certain of the homework problems. We can also talk about the other problems and/or topics from the lectures. On the first day of class we will decide when to hold the problem session. I will also ask you write up and hand in solutions to certain exercises; these will be graded and returned to you. I'll say more about this in class.

Below I list information regarding: this week's hot topics , suggested problems, homework.

Here are suggested problems for each indicated section in Ahlfors. In general you should try to solve all of the problems in Ahlfors, especially if you are planning to take the preliminary examination in Complex Analysis; but, be aware that some of Ahlfors' problems require some ingenuity. As I mentioned above, the other books also have many many good problems, many of which are routine.

• I.1.1, Complex Arithmetic, pp.2-3:1,2
• I.1.2, Square Roots, p.4:3
• I.1.4, Conjugation & Absolute Value, p.9:3,4
• I.1.5, Inequalities, p.11:1,3,4
• I.2.1, Geometric Arithmetic, p.15:1-3
• I.2.2, Roots, p.16:3
• I.2.3, Analytic Geometry, p.17:1-3,5
• I.2.4, Riemann Sphere, p.20:1,2,4,5
• II.3.1-2,Complex Exp & Trig Functions, p.44:1,2,4
• II.3.3-4, Complex Logarithm & Powers, p.47:3-8
• III.1.1-2, Metric Topology, p.53:(1,2),3,4,6,7
• III.1.3, Connectedness, p.58:2,3,6
• III.1.4, Compactness, p.63:2,3,4
• III.1.5, Continuity, p.66:1-5
• II.1.2, Holomorphic Functions, p.28:1-7
• III.2.2, Holomorphic Functions in Region, p.72:1-3
• III.2.3, Conformal Mapping, read pp.73-75
• III.3.1, The Möbius Group, p.78:1-4
• III.3.2, The Cross Ratio, p.80:1-4
• III.3.3, Symmetry, p.82-83:1-8

Here is the assigned homework with due dates. (Here, again is a link to a pdf file for a list of these homework problems). The "Problems Due" are to be handed in during class on the indicated dates. The "For Session" problems will be discussed in that week's problem session.

Due Dates	Problems Due			For Session
21,23 Sept	No Class	First Day	3	No Session
26,28,30 Sept	4	8	10	2,5-8
3,5,7 Oct	13	14	23	12,16-21
10,12,14 Oct	24	27	34	25-33
17,19,21 Oct	35	42	44	1-43
24,26,28 Oct	46	MidTerm	0	44-51
31; 2,4 Nov	59,62 due Fri			Review MT
7,9,11 Nov	65,70 due Wed		Vets Day	55,60,63,69
14,16,18 Nov	68, 73, 79 due Fri			72, 74-78
21,23,25 Nov	81 due Wed		Trky Day	80, 82, 83
28,30; 2 Dec	88 due Fri			1-87
9 Dec	Final Exam 1:30-3:30			Review