COURSE DESCRIPTION: The calculus of complex-valued functions turns out to have unexpected simplicity and power. As an example of simplicity, every complex-differentiable function is automatically infinitely differentiable and equals its Taylor series expansion! As examples of power, the residue calculus permits the computation of difficult integrals, and conformal mapping reduces physical problems on very general domains to problems on the round disc. The easiest proof of the Fundamental Theorem of Algebra, not to mention the first proof of the Prime Number Theorem, used complex analysis. NOTE: this course will move at a good pace and will provide the complex analysis background graduate programs in mathematics desire. We will cover a lot of material and applications. Whenever possible we will prove all results and theorems, either in class or as part of the homework.
Format: lecture. Evaluation will be based primarily on homework, classwork, and exams.
Prerequisites: Either Mathematics 301 or 305 --OR-- permission of instructor. No enrollment limit (expected: 21). 

CONTACTING ME: You can reach me in Bronfman 202 (if I'm there it's office hours), email sjm1@williams.edu, or anonymously through ephsmath@gmail.com (password williams1793;  it used to be the first eight Fibonacci numbers but annoyingly  someone hacked the account and changed it, and Google wouldn't let me restore it).

OBJECTIVES: There are two main goals to this course: to explore complex analysis and see the connections between various subjects, and to learn problem solving skills. We will constantly emphasize the techniques we use to solve problems and prove theorems, as these are applicable to a wide range of problems in the sciences. For a fuller statement as to the objectives of this course, please click here. This includes some fascinating videos with some thought provoking comments about what you should get out of your education.

GRADING / HW: Prepared for Class: 5%, Homework 15%, Midterms 40%, Final 40%. Homework is to be handed in on time, stapled and legible. Late, messy or unstapled homework will not be graded. I encourage you to work in small groups, but everyone must submit their own hw assignment. Extra credit problems should not be included in the general homework, but handed in separately. Very little partial credit is given on these problems. There will be opportunities to do a project and present to the class.

TEXTBOOK: Complex Analysis by Stein and Shakarchi (Princeton University Press, ISBN13: 978-0-691-11385-2). Click here for the introduction, click here for chapter 2.

COURSE NOTES: Below are scanned copies of my lecture notes for the course (click here for additional comments from each day's lecture). Skimming these notes is a good way to prepare for lecture and to review the material; however, it is not the case that everything said in lecture will be in these notes for two reasons: (1) I hope to have a lot of class discussion, and these comments will undoubtably influence which direction we pursue; (2) detailed explanations of many arguments are given in the book, so often I have just jotted down notes to remind myself of what I wanted to mention.

COURSE DISCLAIMER: I may occasionally say things such as `Probability is one of the most useful courses you can take' or 'If you know probability, stats and a programming language then you'll always be able to find employment'. I really should write `you should always be able to find employment', as nothing is certain. Thus, please consider yourself warned and while you may savor the thought of suing me and/or Williams College, be advised against this! I'm saying this because of the recent lawsuit of a graduate who was upset that she didn't have a job, and sued her school!