Mathematics: Theory of Func of Complex Var

Complex functions, differentiation and the Cauchy-Riemann equations, power and Laurent series, integration, calculus of residues, contour integration, isolated singularities, conformal mapping, harmonic functions. Prerequisite: Two semesters of real analysis required.

Prereqs: Two semesters of Real Analysis; Open to Degree and PACE students

After a brief review of real analysis (mainly metric spaces and integration) we will look at basic properties of and results connected with differentiable (“analytic”) functions of a complex variable: the Cauchy-Riemann equations, Cauchy’s Theorem, the Cauchy Integral Formula (several forms, including those involving homotopy), power series expansions and their consequences. We will cover the types of isolated singularities (removable, pole, essential) of analytic functions and the behaviors associated with each (e.g., Casorati-Weierstrass Theorem). We will prove the Maximum Modulus Theorem (more than one version) for analytic functions and some of its consequences (Schwarz’s Lemma). We will study convergence properties of sequences of analytic functions, with special emphasis on normal families. We will study the geometric mapping properties of analytic functions and prove the Riemann Mapping Theorem (RMT). Proving the RMT will bring into play almost everything studied to that point. After that we will treat other topics (e.g., Weierstrass Factorization Theorem, Runge’s Theorem, Hardy spaces) as time permits. The text for the class is John B. Conway’s Functions of One Complex Variable I (2nd edition, Springer). It will be on reserve in the library and for sale in the bookstore, but I recommend buying it online.

Read the assignments, do the homework, come to class. Come to office hours or otherwise contact me if you have questions.

The grade will be based entirely on homework scores. There will be no midterm or final. There will be from 5 to 7 large homework sets, which the students should begin as soon as they get them. Homework will be submitted via Brightspace. I compute grades like this. The average of your homework scores will yield a number between 0 and 100 (possibly higher if I include extra credit problems in any homeworks), which I call your class score. If your class score is 100 or higher, you get an A+; if it's at least 90, the lowest grade you can get is an A-; if it's at least 80, the lowest grade you can get is a B-; and so on. In practice these lower limits often get pushed down.