## John McCarthy, Ph.D.

Operator Theory, One and Several Complex Variables, and Their Interaction

# 5021: Complex Analysis Fall 2021

Instructor           John E. McCarthy
Class                  TuTh 1.00-2.20 in Eads 102

No class October 12 (Fall Break); we will have class November 23.
Office                 105 Cupples I
Office Hours      M 4:00-4:50, Tu 10.00-10.50, Th 2.30-3.20, and by appointment

In office and via Zoom: Meeting ID: 932 2571 1839

Text                    Complex Analysis, Theodore Gamelin

Exams    There will be two exams in the course:

1) Exam 1       Take home, Tuesday October 19
2) Exam 2       Final exam, on Tuesday, December 21, 1-3 pm

Homework

There will be weekly homework sets during the semester, handed out on Thursday and due the following Thursday. Some problems are fairly routine, but many are quite challenging.

Prerequisites

The material of 417-418 and 411-412. Specifically:

Topology: Be familiar with compactness, connectedness. Know what a homotopy is, and what simply connected means.
Analysis: Must be solid on multivariable calculus. Know some theorems about interchanging limits and integrals (e.g. if a sequence of functions converges uniformly).

Content

We will build up the basic theory of analytic functions of a complex variable from scratch, culminating in Cauchy’s theorem and the residue theorem. As time permits, we will cover additional topics. (I intend to cover Gamelin’s entire book over two semesters).

Attendance and Class Participation: 10%

Midterm: 20%

Homework: 30%

Final: 40%

Homework

Homework is the most important part of the course. Whilst talking to other people about it is not dis-allowed, too often this degenerates into one person solving the problem, and other people copying them (often justified to themselves by saying “I provide the ideas, X does the details” – but the details are the key. If you can’t translate the idea into a real proof, you don’t understand the material well enough). So I shall introduce the following rules:
(a) You can only talk to some-one else about a problem if you have made a genuine effort to solve it yourself.
(b) You must write up the solutions on your own. Suspiciously similar write-ups will receive 0 points.
Note, too, that a significant part of this course is learning how to write mathematics. Thus serious attention will be paid to how the solutions are written up. They should be written in full English sentences, and it should be clear what is going on (so, for example, if the argument is lengthy, dividing the proof into well-marked stages is a good idea). Faulty logic will be penalised with negative points. So, if you cannot prove something is true in general, but can do so if you assume an extra hypothesis, you should say so, and state the extra hypothesis you need. This will earn partial credit. But if you claim to prove the result in general, whilst implicitly assuming this hypothesis, you will lose many points, and may end up getting less than zero on a problem. If you really have no idea how to prove something, don’t waffle. If you have an idea that you cannot make into a rigourous proof, say “This is an idea, but I cannot make it into a rigourous proof”.

I do expect you to come to class every day, and to participate in class discussions. I also expect you to stay abreast of the material we are covering, and may call on you at any time to answer a question.

Please feel free to talk to me about the course at any time. My office hours are just times when I will definitely be in my office. I welcome you to stop by whenever you wish. I shall not tell you how to do current homework problems, but if you’ve made some progress, I may give hints.

Class

You must come to class every day, and participate in class discussions, unless you are ill or quarantining. In these events, please email me.

Web Pages

The following web pages may give some interesting sidelights on the material.

The MacTutor History of Mathematics Archive

History of Analysis

Bibliography

The following is a brief bibliography you may find useful. Most books on complex analysis cover the same material, so choose one whose style suits you. The first one listed, by Sarason, is shorter – it is intended to cover one semester.

D. Sarason               Complex Function Theory
L.V. Ahlfors            Complex Analysis: an introduction to the theory of analytic functions of one complex variable
J.B. Conway            Functions of one complex variable
R.E. Greene and S.G. Krantz       Function Theory of one complex variable

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