Instructor John E. McCarthy
Class TuTh 11.30-12.45
Office Hours M: 4-5 Tu: 2.30-3.15 Th: 1.00-1.45 Zoom Office Hours
Phone 935-6753
Modality: This will be decided in consultation with the class. My initial plan is to give in person classes to the
majority of enrolees, and to live stream the classes for those who cannot attend in person. However, once I know who has signed up, I will discuss with the enrolees what works best for them.
Exams There will be two exams in the course:
1) Exam 1 Midterm – Friday March 19.
2) Exam 2 Final exam. Tuesday May 11, 1.00-3.00.
Homework
There will be weekly homework sets during the semester, assigned on Tuesday and due the following Tuesday.
Study/Wellness Days
The following dates are either Study days in the Mathematics and Statistics Department, or Wellness Days in Arts and Sciences:
Tuesday Feb 9
Tuesday Mar 2
Wednesday Mar 3
Monday Mar 22
Monday April 12
On these days, there are no office hours or classes. Any assignment due that day is extended 24 hours
(so the one assigned Feb 2 is due Feb 10, and the one assigned Feb 23 is due Mar 4).
Prerequisites
Math 318, or permission of instructor.
Description
Complex Analysis is an essential tool in (almost) all areas of modern mathematics. It started with
Tartaglia’s solution of the cubic – in order to find the real roots of a real cubic polynomial, the formula
requires complex numbers. The fundamental theorem of algebra says that every complex polynomial
can be factored into linear factors. This means that every matrix has complex eigenvalues, though not necessarily real ones.
Analytic functions of complex variables – functions that can locally be written as power series – are the heart of the subject.
They are both flexible and rigid, in ways we will discuss, and make the subject very attractive.
Content
- Complex Numbers. De Moivre’s Formula
- Complex Differentiation. Cauchy-Riemann equations.
- Harmonic Functions
- Linear fractional transformations
- Exponential and logarithmic functions
- Power series
- Complex Integration
- Cauchy’s theorem
- Some of the many consequences of Cauchy’s theorem – Liouville’s theorem, Maximum modulus theorem, Schwarz’s lemma
- Harmonic functions redux
- Laurent series. Singularities and Poles.
- Residue theorem. Definite integrals.
- Rouche’s theorem. Riemann mapping theorem.
- Homotopy version of Cauchy’s theorem.
Basis for Grading
Attendance and class participation will be 30% of your grade, homework will be 30%, the midterm will be 15%, and the final will be 25%.
Homework
Homework is an extremely 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.
(c) You may not get solutions from the internet.
Academic Integrity
Academic Integrity is especially important when we are struggling to allow remote learning.
During exams, you may not get help from anyone else, or seek solutions online.
I will ask you to write and sign the following Honor Statement on each exam:
“I affirm that I have not given or received any unauthorized help on this assignment, and that this work is my own.”
Class
I expect you to attend class every day, and to participate in class discussions.
It is important to stay abreast of the material. At the beginning of class I will call on people to give definitions or state theorems; this will be part of your grade. I may call on you at any time to answer a question.
Class etiquette (in person): don’t be disruptive or discourteous. No beeping, ringing, crunching, rustling, leaving early or arriving late. No texting, sleeping, checking your phone.
Class etiquette (online): please stay fully engaged – no multitasking. Except for compelling reasons, please keep your video on.
Links
Texts Complex Function Theory by Donald Sarason (AMS, 2007)