Here are notes for my talk “Crash course in complex analysis” for the Gems of TCS seminar.
Abstract: Calculus on the complex numbers has an entirely different attitude from calculus on the reals. The additional structure of the complex numbers greatly enriches the theory of differentiable functions. I will cover (the many variants of) Cauchy’s Theorem, power (Laurent) series, and various inequalities (Hadamard three-lines and three-circles). I’ll give some applications including convergence rates for polynomial approximations and numerical integration, and a recent application to population recovery.