Édouard Goursat
Edouard Goursat
Born	(1858-05-21)21 May 1858 Lanzac,Lot
Died	25 November 1936(1936-11-25) (aged 78) Paris
Alma mater	École Normale Supérieure
Known for	Goursat tetrahedron Goursat theorem Goursat's lemma Inverse function theorem
Scientific career
Fields	Mathematics
Institutions	University of Paris
Doctoral advisor	Jean Gaston Darboux
Doctoral students	Georges Darmois Dumitru Ionescu [ro]

Édouard Jean-Baptiste Goursat (21 May 1858 – 25 November 1936) was a Frenchmathematician, now remembered principally as an expositor for hisCours d'analyse mathématique, which appeared in the first decade of the twentieth century. It set a standard for the high-level teaching ofmathematical analysis, especiallycomplex analysis. This text was reviewed byWilliam Fogg Osgood for the Bulletin of theAmerican Mathematical Society.^[1][2] This led to its translation into English byEarle Raymond Hedrick published by Ginn and Company. Goursat also published texts onpartial differential equations andhypergeometric series.

Edouard Goursat was born inLanzac,Lot. He was a graduate of theÉcole Normale Supérieure, where he later taught and developed hisCours. At that time thetopological foundations of complex analysis were still not clarified, with theJordan curve theorem considered a challenge tomathematical rigour (as it would remain untilL. E. J. Brouwer took in hand the approach fromcombinatorial topology). Goursat's work was considered by his contemporaries, includingG. H. Hardy, to be exemplary in facing up to the difficulties inherent in stating the fundamentalCauchy integral theorem properly. For that reason it is sometimes called theCauchy–Goursat theorem.

Goursat, along withMöbius,Schläfli,Cayley,Riemann,Clifford and others, was one of the 19th century mathematicians who envisioned and explored ageometry of more than three dimensions.^[3]

He was the first to enumerate thefinite groups generated by reflections in four-dimensional space, in 1889.^[4] TheGoursat 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 arefour-dimensional Euclidean polytopes.

He derived a formula for the general displacement in four dimensions preserving the origin, which he recognized as adouble rotation in two completely orthogonal planes.^[5]

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

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

where${\displaystyle \omega }$ is ap-form inn-space andS is thep-dimensional boundary of the (p + 1)-dimensional regionT. Goursat also useddifferential forms to state thePoincaré lemma and its converse, namely, that if${\displaystyle \omega }$ is ap-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.Élie Cartan himself in 1922 gave a counterexample, which provided one of the impulses in the next decade for the development of theDe Rham cohomology of adifferential manifold.

• A Course In Mathematical Analysis Vol I Translated by O. Dunkel and E. R. Hedrick (Ginn and Company, 1904)
• A Course In Mathematical Analysis Vol II, part I Translated by O. Dunkel and E. R. Hedrick (Ginn and Company, 1916) (Complex analysis)
• A Course In Mathematical Analysis Vol II Part II Translated by O. Dunkel and E. R. Hedrick (Ginn and Company, 1917) (Differential Equations)
• Leçons sur l'intégration des équations aux dérivées partielles du premier ordre (Hermann, Paris, 1891)^[6]
• Leçons sur l'intégration des équations aux dérivées partielles du second ordre, à deux variables indépendantes Tome 1 (Hermann, Paris 1896–1898)^[6]
• Leçons sur l'intégration des équations aux dérivées partielles du second ordre, à deux variables indépendantes Tome 2 (Hermann, Paris 1896–1898)^[6]
• Leçons sur les séries hypergéométriques et sur quelques fonctions qui s'y rattachent (Hermann, Paris, 1936–1939)^[7]
• Le problème de Bäcklund (Gauthier-Villars, Paris, 1925)
• Leçons sur le problème de Pfaff (Hermann, Paris, 1922)^[8]
• Théorie des fonctions algébriques et de leurs intégrales : étude des fonctions analytiques sur une surface de Riemann withPaul Appell (Gauthier-Villars, Paris, 1895)^[9]
• Théorie des fonctions algébriques d'une variable et des transcendantes qui s'y rattachent Tome II, Fonctions automorphes with Paul Appell (Gauthier-Villars, 1930)

• Goursat structure
• Goursat's lemma

• Coxeter, H.S.M. (1973).Regular Polytopes (3rd ed.). New York: Dover.
• Katz, Victor (2009).A History of Mathematics: An introduction (3rd ed.). Boston: Addison-Wesley.ISBN 978-0-321-38700-4.

• Media related toÉdouard Goursat at Wikimedia Commons
• O'Connor, John J.;Robertson, Edmund F."Édouard Goursat".MacTutor History of Mathematics Archive.University of St Andrews.
• William Fogg OsgoodA modern French Calculus Bull. Amer. Math. Soc.9, (1903), pp. 547–555.
• William Fogg OsgoodReview: Edouard Goursat, A Course in Mathematical Analysis Bull. Amer. Math. Soc.12, (1906), p. 263.
• Édouard Goursat at theMathematics Genealogy Project

• 1858 births
• 1936 deaths
• 19th-century French mathematicians
• 20th-century French mathematicians
• French mathematical analysts
• Members of the French Academy of Sciences
• École Normale Supérieure alumni
• Academic staff of the University of Paris

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• Short description matches Wikidata
• Use dmy dates from May 2024
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