Poincaré and complex function theory

Gray, Jeremy (2010). Poincaré and complex function theory. In: Bour, Pierre; Rebuschi, Manuel and Rollet, Laurent eds. Construction: Festschrift for Gerhard Heinzmann. Tributes (14). London: College Publications, pp. 3–22.

URL: http://poincare.univ-nancy2.fr/Presentation/?conte...

Abstract

Poincaré is still well known for the mathematical work that first made his name : his discovery in 1880--1881 of automorphic functions. New documents and insights were added in [Gray and Walter 1999], which can also be consulted for references to the well-known history of his work in this area. He is also remembered for one further theorem that grew out of that early work : the uniformisation theorem, which he sketched a proof of in 1883 and then proved rigorously in 1907, as did Koebe independently. The rest of his numerous contributions
to complex function theory are more scattered and do not seem to have been the focus of much attention. In this paper I survey what he did and argue that they tell an eloquent story not only about the state of the subject
in the years around 1900 but about Poincaré's place in the mathematical community of his day. To understand either of these it is necessary to give a quick summary of the prior development of complex function theory, which was growing rapidly into a central topic in all mathematics, and that will occupy the first half of this paper. The second half will consider Poincaré’s contributions. We will see that although he was actively involved in many aspects of the subject, his influence is scarcely to be noticed in the many
books that were published, and I will investigate why that was and what it may tell us about relationship between research and teaching in the years around 1900.

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