Cauchy and the Creation of Complex Function Theory

Between 1814 and 1831, the great French mathematician A. L. Cauchy created practically single-handedly a new branch of pure mathematics. Complex function theory was and remains of central importance, and its creation marked the start of one of the most exciting periods in the development of mathematics. In this book Dr Smithies analyses the process whereby Cauchy created the basic structure of complex analysis, describing first the eighteenth-century background before proceeding to examine the stages of Cauchy's own work, culminating in the proof of the residue theorem and his work on expansions in power series. Smithies describes how Cauchy overcame difficulties including false starts and contradictions brought about by over-ambitious assumptions, as well as the improvements that came about as the subject developed in Cauchy's hands. Controversies associated with the birth of complex function theory are described in detail. Throughout, new light is thrown on Cauchy's thinking during this watershed period. This book makes use of the whole spectrum of available original sources; it will be recognised as the authoritative work on the creation of complex function theory.

Table of Contents:
1. Introduction; 2. The background to Cauchy's work on complex function theory; 3. Cauchy's 1814 memoir on definite integrals; 4. Miscellaneous contributions (1815–1825); 5. The 1825 memoir and associated papers; 6. The calculus of residues; 7. The Lagrange series and the Turin memoirs; 8. Summary and conclusions; References.

Review :
Review of the hardback: 'This exemplary book is the first thorough and comprehensive presentation of Cauchy's creation between 1814 and 1831, of complex function theory … the papers form a diary that testifies to the gradual development of a new field in the mind of an outstanding mathematician.' D. Laugwitz, Darmstadt Review of the hardback: ' … an invaluable guide to anyone interested in the work of a great mathematician'. Jeremy Gray, The Open University Review of the hardback: 'This book is a welcome contribution to the history of mathematical ideas. It is carefully written … in the case of a great mathematician of the first half of the nineteenth century, it is very useful to have a book analyzing his ideas. Smithie's book helps us understand the technical achievements of Cauchy in founding the theory of complex functions.' The Mathematical Gazette Review of the hardback: '… new light is thrown on Cauchy's thinking.' L'Enseignment Mathématique Review of the hardback: '… a unique source not only for the historians of mathematics but to all who are fascinated by the beauty of the complex function theory.' European Mathematical Society