Lecture Analytic combinatorics (Part 2) - Chapter 4: Complex analysis, rational and meromorphic asymptotics

The purpose of this chapter is largely to serve as an accessible introduction or a refresher of basic notions regarding analytic functions. We start by recalling the elementary theory of functions and their singularities in a style tuned to the needs of analytic combinatorics. Cauchy’s integral formula expresses coefficients of analytic functions as contour integrals. Suitable uses of Cauchy’s integral formula then make it possible to estimate such coefficients by suitably selecting an appropriate contour of integration.