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Goursat had a lifelong association with Emile Picard. Julia, who was Goursat's student, later collaborated with him. 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. In 1891 he wrote Lecons sur l'intégration des équations aux dérivées partielles du premier ordre.
Goursat's best known work is Cours d'analyse mathématique (1900-10) which introduced many new analysis concepts. Goursat became a member of the French Academy of Science in 1919.
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