Augustin-Louis Cauchy

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Augustin-Louis Cauchy : biography

df=y September 8 – 23 May 1857
frac oint_C f(z) dz = sum_^n underset{mathrm} f(z), 

where the sum is over all the n poles of f(z) on and within the contour C. These results of Cauchy’s still form the core of complex function theory as it is taught today to physicists and electrical engineers. For quite some time, contemporaries of Cauchy ignored his theory, believing it to be too complicated. Only in the 1840s the theory started to get response, with Pierre-Alphonse Laurent being the first mathematician, besides Cauchy, making a substantial contribution (his Laurent series published in 1843).

Cours d’Analyse

The title page of a textbook by Cauchy. In addition to his work on complex functions, Cauchy was the first to stress the importance of rigor in analysis. His book Cours d’Analyse had a such an impact that Judith Grabiner writes Cauchy was "the man who taught rigorous analysis to all of Europe." This book is frequently noted as being the first place that inequalities, and delta-epsilon arguments were introduced into Calculus. Cauchy exploited infinitesimals and wrote in his introduction that he has been "… unable to dispense with making the principal qualities of infinitely small quantities known…". M. Barany claims that the École mandated the inclusion of infinitesimal methods against Cauchy’s better judgement . Gilain argued that the infinitesimal portions of the book were likely a late insertion. Laugwitz (1989) and Benis-Sinaceur (1973) argued that Cauchy was not forced to teach infinitesimals, pointing out that he continued to use them in his own work as late as 1853…

Cauchy gave an explicit definition of an infinitesimal in terms of a sequence tending to zero. Namely, such a null sequence "becomes" an infinitesimal in Cauchy’s and Lazare Carnot’s terminology. Sources disagree if Cauchy defined his notion of infinitesimal in terms of limits. Some have argued that such a claim is ambiguous, and essentially a play of words on the term "limit". Similarly, some sources contest the claim that Cauchy anticipated Weierstrassian rigor, and point out internal contradictions in post-Weierstrassian Cauchy scholarship relative to Cauchy’s 1853 text on the sum theorem..

BaranyBarany, M. J.: revisiting the introduction to Cauchy’s Cours d’analyse. Historia Mathematica 38 (2011), no. 3, 368–388. http://dx.doi.org/10.1016/j.hm.2010.12.001 recently argued that Cauchy possessed a kinetic notion of limit similar to Newton’s. Regardless of how Cauchy viewed the rigor of using infinitesimal methods, these methods continued in practice long after Cours d’Analyse both by Cauchy and other mathematicians and can be justified by modern techniques.

Taylor’s theorem

He was the first to prove Taylor’s theorem rigorously, establishing his well-known form of the remainder. He wrote a textbookCauchy, Cours d’Analyse de l’École Royale Polytechnique, I.re partie, Analyse Algébrique, Paris (1821) (see the illustration) 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.Cauchy, Leçons sur le Calcul Différentiel, Paris (1829) In spite of these, Cauchy’s own research papers often used intuitive, not rigorous, methods;Morris Kline, Mathematics: The Loss of Certainty, ISBN 0-19-503085-0, p. 176 thus one of his theorems was exposed to a "counter-example" by Abel, later fixed by the introduction of the notion of uniform continuity.

Argument principle, stability

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.

Output

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:

  • (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:

  • (1821)
  • Le Calcul infinitésimal (1823)
  • Leçons sur les applications de calcul infinitésimal; La géométrie (1826–1828)

His other works include:

  • (Paris: Bachelier, 1840–1847)
  • (Imprimerie Royale, 1821)
  • (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).
  • CR Ac ad. Sci. Paris, t. XVII, 449-458 (1843) credited as originating the operational calculus.

Notes