Cauchy Theory

Abstract

In Chapter VII, § 4 we showed how a significant part of the classical theory of holomorphic or analytic functions on C can be obtained from Fourier series. In fact, our universal method for constructing them – a fundamental idea of Cauchy – is to integrate holomorphic functions along curves drawn in their domains of definition and thereby obtain a version of the “ fundamental theorem of differential and integral calculus ” (FT) for holomorphic functions, and then to deduce countless consequences.