Edouard Goursat
Encyclopedia
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
. This led to its translation in English by Earle Raymond Hedrick
published by Ginn and Company. Goursat also published texts on partial differential equation
s and hypergeometric series
.
, 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.
where
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 
is a p-form, then 
if and only if there is a (p − 1)-form 
with
. However Goursat did not notice that the "only if" part of the result depends on the domain of 
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.
French people
The French are a nation that share a common French culture and speak the French language as a mother tongue. Historically, the French population are descended from peoples of Celtic, Latin and Germanic origin, and are today a mixture of several ethnic groups...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, 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
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, especially complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
. This text was reviewed by William Fogg Osgood
William Fogg Osgood
William Fogg Osgood was an American mathematician, born in Boston.In 1886, he graduated from Harvard, where, after studying at the universities of Göttingen and Erlangen , he was instructor , assistant professor , and thenceforth professor of mathematics...
for the Bulletin of the American Mathematical Society
American Mathematical Society
The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
. This led to its translation in English by Earle Raymond Hedrick
Earle Raymond Hedrick
Earle Raymond Hedrick , was an American mathematician and a vice-president of the University of California.Hedrick was born in Union City, Indiana....
published by Ginn and Company. Goursat also published texts on partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s and hypergeometric series
Hypergeometric series
In mathematics, a generalized hypergeometric series is a series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by...
.
Life
Edouard Goursat was born in LanzacLanzac
Lanzac is a commune in the Lot department in south-western France....
, Lot
Lot (département)
Lot is a department in the southwest of France named after the Lot River.- History :Lot is one of the original 83 departments created during the French Revolution on March 4, 1790. It was created from part of the province of Languedoc. In 1808, some of the original southeastern cantons were...
. He was a graduate of the École Normale Supérieure
École Normale Supérieure
The École normale supérieure is one of the most prestigious French grandes écoles...
, 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
Jordan curve theorem
In topology, a Jordan curve is a non-self-intersecting continuous loop in the plane, and another name for a Jordan curve is a "simple closed curve"...
considered a challenge to mathematical rigour (as it would remain until L. E. J. Brouwer took in hand the approach from combinatorial topology
Combinatorial topology
In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces were regarded as derived from combinatorial decompositions such as simplicial complexes...
). Goursat’s work was considered by his contemporaries, including G. H. Hardy
G. H. Hardy
Godfrey Harold “G. H.” Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis....
, 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 formwhere
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 is a p-form, then if and only if there is a (p − 1)-form with. However Goursat did not notice that the "only if" part of the result depends on the domain of 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 cohomologyDe Rham cohomology
In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes...
of a differential manifold.
Books by Edouard Goursat
- 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)
- Leçons sur l'intégration des équations aux dérivées partielles du second ordre, à deux variables indépendantes Tome 1 (Hermann, Paris 1896–1898)
- Leçons sur l'intégration des équations aux dérivées partielles du second ordre, à deux variables indépendantes Tome 2 (Hermann, Paris 1896–1898)
- Leçons sur les séries hypergéométriques et sur quelques fonctions qui s'y rattachent (Hermann, Paris, 1936–1939)
- Le problème de Bäcklund (Gauthier-Villars, Paris, 1925)
- Leçons sur le problème de Pfaff (Hermann,Paris, 1922)
- Théorie des fonctions algébriques et de leurs intégrales : étude des fonctions analytiques sur une surface de Riemann with Paul Appell (Gauthier-Villars, Paris, 1895)
- Théorie des fonctions algébriques d'une variable et des transcendantes qui s'y rattachent Tome II, Fonctions automorphes with Paul Appell (Gauthier-Villars, 1930)
External links
- William Fogg Osgood A modern French Calculus Bull. Amer. Math. Soc. 9, (1903), pp. 547–555.
- William Fogg Osgood Review: Edouard Goursat, A Course in Mathematical Analysis Bull. Amer. Math. Soc. 12, (1906), p. 263.