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From Wikipedia, the free encyclopedia,
http://en.wikipedia.org/wiki/Augustin-Louis_Cauchy
Baron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors. He defined continuity in terms of infinitesimals and gave several important theorems in complex analysis and initiated the study of permutation groups in abstract algebra. A profound mathematician, Cauchy exercised a great influence over his contemporaries and successors. His writings cover the entire range of mathematics and mathematical physics.
Cauchy was a prolific writer; he wrote approximately eight hundred research articles and five complete textbooks. He was a devout Roman Catholic, strict Bourbon royalist, and a close associate of the Jesuit order.
In the theory of light he worked on Fresnel's wave theory and on the dispersion and polarization of light. He also contributed significant research in mechanics, substituting the notion of the continuity of geometrical displacements for the principle of the continuity of matter. He wrote on the equilibrium of rods and elastic membranes and on waves in elastic media. He introduced a 3 x 3 symmetric matrix of numbers that is now known as the Cauchy stress tensor. In elasticity, he originated the theory of stress, and his results are nearly as valuable as those of Simeon Poisson. Other significant contributions include being the first to prove the Fermat polygonal number theorem.
Cauchy is most famous for his single-handed development of complex function theory.
He was the first to prove Taylor's theorem rigorously, establishing his well-known form of the remainder. He wrote a textbook for his students at the École Polytechnique in which he developed the basic theorems of mathematical analysis as rigorously as possible. In this book he gave the necessary and sufficient condition for the existence of a limit in the form that is still taught. Also Cauchy's well-known test for absolute convergence stems from this book: Cauchy condensation test. In 1829 he defined for the first time a complex function of a complex variable in another textbook. In spite of these, Cauchy's own research papers often used intuitive, not rigorous, methods; thus one of his theorems was exposed to a "counter-example" by Abel, later fixed by the introduction of the notion of uniform continuity.
In a paper published in 1855, two years before Cauchy's death, he discussed some theorems, one of which is similar to the "Argument Principle" in many modern textbooks on complex analysis. In modern control theory textbooks, the Cauchy argument principle is quite frequently used to derive the Nyquist stability criterion, which can be used to predict the stability of negative feedback amplifier and negative feedback control systems. Thus Cauchy's work has a strong impact on both pure mathematics and practical engineering.
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Cauchy was very productive, in number of papers second only to Leonhard Euler. It took almost a century to collect all his writings into 27 large volumes:
Oeuvres complètes d'Augustin Cauchy publiées sous la direction scientifique de l'Académie des sciences et sous les auspices de M. le ministre de l'Instruction publique (27 volumes) (Paris : Gauthier-Villars et fils, 1882–1974)
His greatest contributions to mathematical science are enveloped in the rigorous methods which he introduced; these are mainly embodied in his three great treatises:
Cours d'analyse de l'École royale polytechnique (1821)
Le Calcul infinitésimal (1823)
Leçons sur les applications de calcul infinitésimal; La géométrie (1826–1828)
His other works include:
Exercices d'analyse et de physique mathematique (Volume 1)
Exercices d'analyse et de physique mathematique (Volume 2)
Exercices d'analyse et de physique mathematique (Volume 3)
Exercices d'analyse et de physique mathematique (Volume 4) (Paris: Bachelier, 1840–1847)
Analyse algèbrique (Imprimerie Royale, 1821)
Nouveaux exercices de mathématiques (Paris : Gauthier-Villars, 1895)
Courses of mechanics (for the École Polytechnique)
Higher algebra (for the Faculté des Sciences)
Mathematical physics (for the Collège de France)
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