I.1. During the years 1814–1831 Cauchy created the basic framework of complex function theory. The aim of the present work is to trace the details of this process, beginning with his 1814 memoir on definite integrals, and taking the story as far as the two Turin memoirs of 1831. That year marks a major break in the development of his work in this direction: it is true that for some time thereafter he was making important applications of his results, but he made no further substantial contributions to the central core of the theory until after his return to teaching in 1849.

The available evidence for Cauchy's ideas lies in his published work, practically all his manuscripts and correspondence having disappeared. I hope to show that a connected story emerges from a careful study of his publications, and that we can trace a gradual development in Cauchy's ideas about complex function theory through a number of stages of his work.

We shall concentrate our attention on the main line of development, and we shall leave a number of topics undiscussed, including, for instance, his work on Fourier transforms and differential equations, some of which does involve applications of his results on complex function theory. Important as these achievements were in their own right, they do not seem to have had any significant influence on his development of the central framework of the theory.