Math 422 Introduction to Complex Analysis

Spring 2016, UAF

Instructor: Ed Bueler

Office: Chapman 301C.
Phone: 474-7693
eMail: elbueler@alaska.edu

Class Times & Room: MWF 9:15--10:15 Bunnell 410
CRN: 34426
Text: Gamelin, Complex Analysis, Corrected ed., Springer 2003

Course Website: bueler.github.io/M422S16/index.html

Ed's schedule and Office Hours: bueler.github.io/OffHrs.htm

Course Content and Topics:

The catalog gives this terse description of the course: Complex functions including series, integrals, residues, conformal mapping and applications.

In more detail, we will cover these topics:

complex numbers and their (easy) algebra
analytic (= complex-differentiable) functions
Cauchy-Riemann equations and harmonic functions
mean value property and maximum principle
conformal mappings
major theorems: Cauchy's, Liouville's, Goursat's, and Morera's
Cauchy integral formula
power series
residues

While we will cover less than half of it, the textbook has good explanations and exercises. (And it is not expensive!) The second half of the textbook, which we will not use, covers material for a more advanced course, but it also shows some more fascinating applications of complex analysis. Note that during the course I will bring up applications as they seem appropriate, but the content here is not primarily "applied mathematics".

Goals and Outcomes:

Complex analysis is an essential tool both for applications of mathematics and for understanding further theory. The goals of this class are that you

gain a clearer and deeper understanding of algebra (e.g. polynomials) and calculus (e.g. limits and power series), as they might appear in high school and college courses in which you might be the teacher,
possess a toolbox useful for applications (e.g. power series, residue calculus, conformal maps), including the differential equations of mathematical physics and engineering, and
begin to understand the topology and geometry of two-dimensional manifolds (e.g. through seeing the Riemann surfaces of multi-valued functions), for learning more advanced mathematics and theoretical physics.

Assigned Work, and How Your Grade is Built:

Weekly homework forms 45% of your score for the class, and is thus essential to your grade. It is equally essential to preparing for the exams. The homework will include by-hand computations and proofs. My expectations for proof style are modest, as I know that Math 265 "Introduction to Mathematical Proofs" is not a prerequisite to this course. The homework assignments themselves, and their due dates, will be posted at the Course Website bueler.github.io/M422S16/index.html.

There will be two in-class Midterm Exams and one primarily in-class Final Exam. In fact a small part of the Final Exam will be take-home, but the majority in-class.

In summary:

Work
Homework
Midterm Exam I
Midterm Exam II
Final Exam

Percent of Grade
45%
15%
15%
25%

Dates
weekly
in class, Friday 19 February
in class, Friday 1 April
in class, Thursday 5 May, 8:00--10:00am

Based on your raw homework and exam scores, I guarantee grades according to the following schedule:
90 - 100 % = A, 79 - 89 % = B, 68 - 78 % = C, 57 - 67 % = D, 0 - 56 % = F.
This schedule is a guarantee. I reserve the right to increase your grade above this schedule based on the actual difficulty of the work and/or upon average class performance.

Policies:

The Dept of Mathematics and Statistics has reasonable policies on incompletes, late withdrawals, early final examinations, etc.; see www.uaf.edu/dms/policies. You are covered by the UAF Student Code of Conduct. I will work with the Office of Disabilities Services (208 WHIT, 474-5655) to provide reasonable accommodation to student with disabilities.

Prerequisites:

Officially, the only prerequisite is Math 302 Differential Equations. In actuality, Math 422 course uses a great deal of material from Calculus II and III (Math 252,253), but these courses are prerequisites to Math 302.

The catalog also says, accurately, that: May be taken independently of MATH F421.

If you have taken Math 302 or equivalent then this is a good course for you. It is probably true that those students who have taken more 300-level Math courses will have an easier time in this course, on average, than those who have taken fewer. This is what math professors call the effect of having more "mathematical maturity." It is a nebulous thing, really ...