Goursat had an inspired collection of teachers at École Normale Supérieure and they added to the encouragement given by Picard. His teachers included Jean Darboux and Charles Hermite who influenced him to work on analysis and its applications. He later wrote (in 1935) :-
Hermite is the first who revealed to me the artistic side of mathematics.Darboux certainly recognise the huge potential in his student and wrote in 1879:-
The student Goursat whose development was extremely rapid, is an excellent mathematician, sure to become as superior a teacher as Paul Appell and Émile Picard.But others such as Claude Bouquet and Charles Briot also provided role models that Goursat assimilated into the teaching style that he developed. He began teaching at the University of Paris in 1879, receiving his doctorate in 1881 from l'École Normale Supérieure for his thesis Sur l'equation différentialle linéaire qui admet pour intégrale la série hypergéometrique .
He then taught in Toulouse until 1885. The next 12 years were spent back at École Normale Supérieure where his lectures would form the basis of his famous three volume analysis text. Then he taught analysis at the University of Paris until his retirement.
The Cauchy-Goursat theorem states the integral of a function round a simple closed contour is zero if the function is analytic inside the contour. Cauchy had established the theorem with the added condition that the derivative of the function was continuous. Goursat removed this extra condition in Démonstration du théorèm de Cauchy (1884). He then produced an impressive series of papers which contributed to almost every area of analysis which was being studied at that time. Not only was his work broad, but it was remarkable for its depth :-
Goursat's papers on the theory of linear differential equations and their rational transformations, as well as his studies on hypergeometric series, Kummer's equation, and the reduction of abelian integrals form, in the words of Picard "a remarkable ensemble of works evolving naturally one from the other". Goursat introduced the notion of orthogonal kernels and semiorthogonals in connection with Erik Fredholm's work on integral equations.In  Katz notes that it was Goursat who first noted the generalized Stokes theorem. He also remarks that Goursat used differential forms to state and prove Poincaré's lemma for arbitrary order forms and its converse.
In 1891 Goursat wrote Leçons sur l'intégration des équations aux dérivées partielles du premier ordre . However, his best known work is Cours d'analyse mathématique (1902-13) which introduced many new analysis concepts. The Publisher writes:-
Édouard Goursat's three-volume 'A Course in Mathematical Analysis' remains a classic study and a thorough treatment of the fundamentals of calculus. As an advanced text for students with one year of calculus, it offers an exceptionally lucid exposition. Volume 1 covers applications to geometry, expansion in series, definite integrals, and derivatives and differentials. Volume 2 explores functions of a complex variable and differential equations. Volume 3 surveys variations of solutions and partial differential equations of the second order and integral equations and calculus of variations.It is almost certain that l'Hôpital's rule, for finding the limit of a rational function whose numerator and denominator tend to zero at a point, is so named because Goursat named the rule after de l'Hôpital in his Cours d'analyse mathématique . Certainly the rule appears in earlier texts (for example it appears in the work of Euler), but Goursat is the first to attach de l'Hôpital's name to it.
Despite working on the many new editions of Cours d'analyse mathématique , Goursat found time to write other texts such as Le problème de Backlund (1925), and Leçons sur les séries hypergéométriques et sur quelles fonctions qui s'y rattachent (1936).
Julia, who was Goursat's student, later collaborated with him. He said :-
... in the name of all those who received ... not only the treasures of your science, but also the treasures of your heart, let me express ... our faithful gratitude ... having received from you the nourishment of the soul, the breadth of science and the example of virtue.Goursat received many honours for his outstanding contributions. He received the Grand Prix des Sciences Mathématique in 1886, the Prix Poncelet in 1889, and the Prix Petit d'Ormoy in 1891. He was elected to the Academy of Sciences in Paris in 1919, became Chevalier de la Légion d'Honneur, and was elected president of the French Mathematical Society (Société Mathématique de France) in 1895.
Article by: J J O'Connor and E F Robertson