Overview

Description

CALENDAR DESCRIPTION:

Conformal mapping, Cauchy Integral Formula, Analytic Continuation, Riemann Mapping Theorem, Argument Principle. Quantitative.

COURSE DETAILS:

Who should take this course? The course is aimed at students who are comfortable with being pushed outside their mathematical comfort zone. Complex analysis is a field whose knowledge is essential for all mathematicians, since the field is, in some sense, the grandmomma of all of mathematics. So if you’d like to see some beautiful mathematics and its connections to a lot of other fields, this is the course for you.

What is the material we’ll see in this class? We’ll briskly review some material on analytic functions, the Cauchy-Riemann equations, harmonic functions, the Cauchy integral formula and residue calculus. We’ll examine some consequences of the Cauchy integral formula, including the Open Mapping Theorem, which is a central result in functional analysis. We’ll then revisit singularities and the ’walking the dog’ theorem. We’ll spend time on the properties of mappings by elementary functions, conformal mappings and the Riemann mapping theorem.

We’ll see the elegant connection between the study of complex variables and certain partial differential equations in the plane. We’ll see how simply periodic functions lead to Fourier theory, and doubly periodic functions lead to the Weierstrass-P- function and the Riemann-Zeta function, and from there to number theory.

You’ve already seen at least two terms of analysis before, and know the drill: the key to success in this course is practice.

Grading

Materials

REQUIRED READING:

RECOMMENDED READING: