Cours d'analyse de l'École polytechnique

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Cours d'analyse de l'Ècole polytechnique is an 1882 book on mathematical analysis by Camille Jordan. The three volumes (tomes) address differential calculus, integral calculus, and differential equations. The book is remembered for introducing Jordan content of sets in a plane.

Tome II was published in 1883, and tome III in 1897. A second edition was started in 1893 and completed in 1896.

Contents

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  • Tome I: variables réelles, variables complexes, séries, applications géométriques de la série de Taylor, courbes planes algébraiques.
  • Tome II : intégrales indéfinies, intégrales définies, fonctions représentées par des intégrales définies, potentials Newtoniens, séries de Fourier, intégrals complexes, fonctions elliptiques, intégrales Abéliennes.
  • Tome III : equations differentielles ordinaires, equations aux derivées partiellles, equationes linéaires, calcul des variations.

Notable developments

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Beyond the service as a textbook for students at École polytechnique, Jordan's book advanced mathematical science as follows: Jordan enunciated and proved a topological property of a simple, closed, non-self-intersecting plane curve: the Jordan curve theorem. In the question of which real-valued functions have an integral, researchers sought the appropriate refinement: "Historically it was the introduction of the concept of measurability in the work of Peano and Jordan that was to suggest the manner of insuring the requisite refinement." [1] See Jordan content. In the study of convergence of Fourier series, Jordan extended a criterion first stated by Johann Dirichlet: the Dirichlet–Jordan test.

Mentions

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"In the 1880s and 1890s the most significant and influential theory of the integral was Camille Jordan's."[2]

[The Cours] of Jordan occupies amongst [Treatises on Analysis] a pre-eminent place, for aesthetic reasons on one hand, but also because, if it constitutes an admirable setting out of the results of classical analysis, it announces in many ways modern analysis and prepares the way for it.[3]

Further reading

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  • Henri Lebesgue (1915) "Review: Third edition of volume II of Cours de analyse de l’Ecole Polytechnique ", Bull. des Sci. Math. 38: 18

References

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  1. ^ Thomas W. Hawkins Jr. (1970) Lebesgue’s Theory of Integration: Its Origins and Development, pages 70, University of Wisconsin Press ISBN 0-299-05550-7
  2. ^ J. Ferreires & J. J. Gray (2006) The Architecture of Modern Mathematics: essays in history and philosophy, page 379, Oxford University Press ISBN 0-19-856793-6
  3. ^ Nicolas Bourbaki, translator John Meldrum (1994) Elements of the History of Mathematics, page 198, ISBN 3-540-19376-6
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