# Édouard Goursat

Last updated on 23 July 2017

Édouard Jean-Baptiste Goursat (21 May 1858 – 25 November 1936) was a French mathematician, now remembered principally as an expositor for his Cours d'analyse mathématique, which appeared in the first decade of the twentieth century. It set a standard for the high-level teaching of mathematical analysis, especially complex analysis. This text was reviewed by William Fogg Osgood for the Bulletin of the American Mathematical Society.[1][2] This led to its translation in English by Earle Raymond Hedrick published by Ginn and Company. Goursat also published texts on partial differential equations and hypergeometric series.

Goursat Edouard.jpg

## Life

Edouard Goursat was born in Lanzac, Lot. He was a graduate of the École Normale Supérieure, where he later taught and developed his Cours. At that time the topological foundations of complex analysis were still not clarified, with the Jordan curve theorem considered a challenge to mathematical rigour (as it would remain until L. E. J. Brouwer took in hand the approach from combinatorial topology). Goursat’s work was considered by his contemporaries, including G. H. Hardy, to be exemplary in facing up to the difficulties inherent in stating the fundamental Cauchy integral theorem properly. For that reason it is sometimes called the Cauchy–Goursat theorem.

## Work

Goursat was the first to note that the generalized Stokes theorem can be written in the simple form

${\displaystyle \int _{S}\omega =\int _{T}d\omega }$

where ${\displaystyle \omega }$ is a p-form in n-space and S is the p-dimensional boundary of the (p + 1)-dimensional region T. Goursat also used differential forms to state the Poincaré lemma and its converse, namely, that if ${\displaystyle \omega }$ is a p-form, then ${\displaystyle d\omega =0}$ if and only if there is a (p − 1)-form ${\displaystyle \eta }$ with ${\displaystyle d\eta =\omega }$. However Goursat did not notice that the "only if" part of the result depends on the domain of ${\displaystyle \omega }$ and is not true in general. E. Cartan himself in 1922 gave a counterexample, which provided one of the impulses in the next decade for the development of the De Rham cohomology of a differential manifold.