Édouard Goursat

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.

Goursat, along with Möbius, Schläfli, Cayley, Riemann, Clifford and others, was one of the 19th century mathematicians who envisioned and explored a geometry of more than three dimensions.{{Cite journal|last=Stillwell|first=John|author-link=W:John Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}}

He was the first to enumerate the finite groups generated by reflections in four-dimensional space, in 1889.{{Sfn|Coxeter|1973|p=209|loc=§11.x}} The Goursat tetrahedra are the fundamental domains which generate, by repeated reflections of their faces, uniform polyhedra and their honeycombs which fill three-dimensional space. Goursat recognized that the honeycombs are four-dimensional Euclidean polytopes.

He derived a formula for the general displacement in four dimensions preserving the origin, which he recognized as a double rotation in two completely orthogonal planes.{{Sfn|Coxeter|1973|p=216|loc=§12.1 Orthogonal transformations}}

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

:$\backslash int\_S\; \backslash omega\; =\; \backslash int\_T\; d\; \backslash omega$

where $\backslash omega$ 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 $\backslash omega$ is a p-form, then $d\backslash omega=0$ if and only if there is a (p − 1)-form $\backslash eta$ with

$d\; \backslash eta=\backslash omega$. However Goursat did not notice that the "only if" part of the result depends on the domain of $\backslash omega$ and is not true in general. Élie 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.

• [https://archive.org/details/coursemathanalys01gourrich A Course In Mathematical Analysis Vol I] Translated by O. Dunkel and E. R. Hedrick (Ginn and Company, 1904)
• [https://archive.org/details/coursemathema0102gourrich A Course In Mathematical Analysis Vol II, part I] Translated by O. Dunkel and E. R. Hedrick (Ginn and Company, 1916) (Complex analysis)
• [https://archive.org/details/differentalequat033197mbp A Course In Mathematical Analysis Vol II Part II] Translated by O. Dunkel and E. R. Hedrick (Ginn and Company, 1917) (Differential Equations)
• [http://name.umdl.umich.edu/ACR1803.0001.001 Leçons sur l'intégration des équations aux dérivées partielles du premier ordre] (Hermann, Paris, 1891){{cite journal|author=Lovett, Edgar Odell|authorlink=Edgar Odell Lovett|title=Review: Goursat's Partial Differential Equations|journal=Bull. Amer. Math. Soc.|year=1898|volume=4|issue=9|pages=452–487|url=http://www.ams.org/journals/bull/1898-04-09/S0002-9904-1898-00540-2/|doi=10.1090/S0002-9904-1898-00540-2|doi-access=free}}
• [http://gallica.bnf.fr/document?O=N084146 Leçons sur l'intégration des équations aux dérivées partielles du second ordre, à deux variables indépendantes Tome 1]{{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }} (Hermann, Paris 1896–1898)
• [http://gallica.bnf.fr/document?O=N084147 Leçons sur l'intégration des équations aux dérivées partielles du second ordre, à deux variables indépendantes Tome 2]{{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }} (Hermann, Paris 1896–1898)
• [http://gallica.bnf.fr/document?O=N038309 Leçons sur les séries hypergéométriques et sur quelques fonctions qui s'y rattachent]{{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }} (Hermann, Paris, 1936–1939){{cite journal|author=Szegő, G.|authorlink=Gábor Szegő|title=Review: Leçons sur les séries hypergéométriques et sur quelques fonctions qui s'y rattachent by É. Goursat|journal=Bull. Amer. Math. Soc.|year=1938|volume=44|issue=1, Part 1|pages=16–17|url=http://www.ams.org/journals/bull/1938-44-01/S0002-9904-1938-06654-2/S0002-9904-1938-06654-2.pdf|doi=10.1090/s0002-9904-1938-06652-9|doi-access=free}}
• [http://gallica.bnf.fr/document?O=N038954 Le problème de Bäcklund]{{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }} (Gauthier-Villars, Paris, 1925)
• [http://gallica.bnf.fr/document?O=N099552 Leçons sur le problème de Pfaff]{{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }} (Hermann, Paris, 1922){{cite journal|author=Dresden, Arnold|authorlink=Arnold Dresden|title=Review: Leçons sur le problème de Pfaff|journal=Bull. Amer. Math. Soc.|year=1924|volume=30|issue=7|pages=359–362|url=http://www.ams.org/journals/bull/1924-30-07/S0002-9904-1924-03903-2/|doi=10.1090/s0002-9904-1924-03903-2|doi-access=free}}
• [http://gallica.bnf.fr/document?O=N099595 Théorie des fonctions algébriques et de leurs intégrales : étude des fonctions analytiques sur une surface de Riemann]{{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }} with Paul Appell (Gauthier-Villars, Paris, 1895){{cite journal|author=Osgood, W. F.|title=Review: Théorie des fonctions algébriques et de leurs intégrales, by P. Appell and É. Goursat|journal=Bull. Amer. Math. Soc.|year=1896|volume=2|issue=10|pages=317–327|url=http://www.ams.org/journals/bull/1896-02-10/S0002-9904-1896-00353-0/|doi=10.1090/s0002-9904-1896-00353-0|doi-access=free}}
• [http://gallica.bnf.fr/document?O=N092706 Théorie des fonctions algébriques d'une variable et des transcendantes qui s'y rattachent Tome II, Fonctions automorphes]{{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }} with Paul Appell (Gauthier-Villars, 1930)

• Goursat problem
• Goursat structure
• Goursat's lemma

{{reflist}}

• {{Cite book | last=Coxeter | first=H.S.M. | author-link=Harold Scott MacDonald Coxeter | year=1973 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=Regular Polytopes (book) }}
• {{Cite book|first=Victor |last=Katz |title=A History of Mathematics: An introduction |edition=3rd |publisher=Addison-Wesley |location=Boston |year=2009 |isbn=978-0-321-38700-4 }}

• {{Commons category-inline}}
• {{MacTutor Biography|id=Goursat}}
• William Fogg Osgood [http://projecteuclid.org/euclid.bams/1183417526 A modern French Calculus] Bull. Amer. Math. Soc. 9, (1903), pp. 547–555.
• William Fogg Osgood [http://projecteuclid.org/euclid.bams/1183418774 Review: Edouard Goursat, A Course in Mathematical Analysis] Bull. Amer. Math. Soc. 12, (1906), p. 263.
• {{MathGenealogy |id=96283}}

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Category:1858 births

Category:1936 deaths

Category:19th-century French mathematicians

Category:20th-century French mathematicians

Category:French mathematical analysts

Category:Members of the French Academy of Sciences

Category:École Normale Supérieure alumni

Category:Academic staff of the University of Paris