Applied Complex Analysis
SLN 10218
CMU 228, MWF 1:30-2:20pm
Prereqs: Amath 401 or equivalent , or Instructor Permission
Instructor: Bernard Deconinck
deconinc@uw.edu
Tel: 206-543-6069
Office Hours: M 10-11, T9-11
Course Description
Complex variable and associated topics. Branch cuts, series and product expansions. Contour integration, numerical implications. Harmonic functions. Complex potential (and singularities) in physical problems. Conformal mapping; applications and examples. Fourier and Laplace transforms and applications.Textbook
The required textbook for this course is Complex Variables: Introduction and Applications (Cambridge Texts in Applied Mathematics) by Mark Ablowitz and Athanassios Fokas. This books covers all the standard topics for a first graduate-level course on complex variables, from an applied point of view. Other excellent books on complex variables are (in alphabetical order): Functions of a Complex Variable (SIAM publishing), by George Carrier, Max Krook, and Carl E. Pearson (an outstanding applied text, by a member of our department), Function Theory of One Complex Variable (AMS Publishing), by Robert Greene and Steven Krantz, Analytic Function Theory, vols. 1 and 2 (Chelsea Publishing), by Einar Hille (a classic with a wealth of information), Theory of functions of a complex variable (Chelsea Publishing) by Alexsei I. Markushevich (a colossal work (1180pp!), and a great reference to have around).Hand-written lecture notes are available. I will put in an effort to start typing these up. We'll see how far I get. It might depend on the amount of new material I include.
Course Canvas Page
I will use Canvas to post homework sets, link to the class message board, etc. You will need a UW account and be enrolled in the course to access this page: https://canvas.uw.edu/courses/986735
Syllabus
- Complex numbers and elementary functions (complex numbers, properties, stereographic projection, elementary functions, limits, continuity, differentiation)
- Analytic functions and integration (Cauchy-Riemann equations, multivalued functions, Riemann surfaces, integration, Cauchy's theorem, Cauchy's formula, generalizations, Liouville's theorem, Morera's theorem)
- Sequences, series and singularities (definitions, Taylor series, Laurent series, singularities, analytic continuation, infinite products)
- Residue calculus and applications (The residue theorem, definite integrals, principal-value integrals, integrals with branch points, the argument principle, Fourier and Laplace transforms)
- As time permits, a special topic chosen to suit class interest (possibilities: conformal mappings, water waves, elliptic functions, the prime number theorem, etc)
Learning Objectives and Instructor Expectations
Complex analysis, in my opinion, is one of the most beautiful areas of mathematics. The good news for us is that it is widely applicable in just about any field.But there is bad news: just like any other advanced area it requires work. To facilitate your investigations of the course materials, there will be weekly homework sets, which will test your stamina and your inherently peaceful nature. Further, there will be an in-class midterm and final exam. I will be available to help you with any and all questions you have regarding this class. Please take advantage of this available help, both in class by participating, and out of class, by coming to office hours and asking questions.
Grading
Homework sets are assigned weekly. Homework is due at the beginning of class on its due date. Late homework is not accepted, as homework solutions are posted immediately after class. Every homework set you hand in should have a header containing your name, student number, due date, course, and the homework number as a title. Your homework should be neat and readable. The TA is very much allowed to subtract points for presentation.
