The Laurent Expansion Without Cauchy's Integral Theorem

Since Cauchy's time the theory of analytic functions of a complex variable has depended on complex integration theory, and in particular on the fundamental integral theorem (1825) and integral formulas bearing his name. Cauchy defined an analytic function to be one which had a continuous first derivative in a region D, and showed that an analytic function had derivatives of all orders in D. It was not until 1900, with E. Goursat's famous proof of Cauchy's integral theorem, that the continuity of the first derivative could be inferred from its mere existence at all points of D.