Help support MIT OpenCourseWare by shopping at Amazon.com! MIT OpenCourseWare offers direct links to Amazon.com to purchase the books cited in this course. Click on the Amazon logo to the left of any citation and purchase the book from Amazon.com, and MIT OpenCourseWare will receive up to 10% of all purchases you make. Your support will enable MIT to continue offering open access to MIT courses.

Course Objective

The aim of the course is to teach the principal techniques and methods of analytic function theory. This is quite different from real analysis and has much more geometric emphasis. It also has significant applications to other fields like analytic number theory.

Prerequisite

Analysis I (18.100B) or the equivalent.

Text

Ahlfors, Lars V. Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable. 3rd ed. New York, NY: McGraw-Hill, 1979. ISBN: 9780070006577.

This is a text with an attractive geometric flavor. I plan to cover most of the material up to p. 232, but will add two lectures devoted to a complete proof of the Prime Number Theorem, which fits very naturally in with the text material on the Riemann zeta function.

In most of the lectures I will add some material not in the text, partly giving alternative proofs, and partly working through solutions of some of the more interesting problems. In Lec #2, #13, #14, #16, #19, #21, and #22 the treatment is really quite different from the corresponding material in the text. Lec #21 and #22 actually contain a complete proof of the Prime Number Theorem.

Since the lectures deviate so much from the text, regular attendance is strongly recommended.

Homework Assignments

Given out every other Thursday, due the next Thursday. This will be selected from problems in the text.

Exams

Two in-class tests following Lec #4 and #16. Also a final examination.

Grading

The final grade is based on a cumulative point total.

General Remarks

Working on the homework problems is very important for your understanding and command of the subject. Be sure you allow plenty of time for this. It is ok for you to discuss the problems with another student but you must give your own exposition of the solution. In the process you will often be lead to a better solution. It is very important to do this exposition in a neat fashion. This is the right place for you to develop a clear mathematical writing style. (Tex is of course most welcome). Sloppy homework might not be accepted.

There are dozens of books available on the topic of complex variables. I urge you strongly to browse in some of these in the library. Caratheodory's books Theory of Functions Vol. I and II are closely related to our text. Also, I would particularly recommend Polya and Szegö: Problems and Theorems in Analysis. Several chapters there deal with the subject of complex variables. Rudin's book, Real and Complex Analysis is also a valuable reference.