From the Theory of Singular Integrals to the Calculus of Residues

Abstract

Cauchy’s crowning achievement in analysis was unquestionably his theory of complex functions, a branch of mathematics that, except for some interruptions, commanded his attention from 1814 until his death. Few mathematicians concerned themselves with Cauchy’s theory before the late 1840s. Almost all of the progress made in this area until that time was due to Cauchy. The principal lines followed by his work are well known. Cauchy first embarked on what would become his complex function theory by way of studying the integration along closed paths in the complex plane; he did not undertake to investigate analytic functions of a complex variable until 1831 and only much later, in 1846, did he begin to work out the fundamental notions that would govern his complex function theory.