Malmsten's Proof of the Integral Theorem

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[6]. However, the integration of the partial fractions resulted in logarithmsof complex numbers, which gave rise to some problems. Eventually, theseproblems were solved by the well-known formula of Euler, namely;log z = log r + i(ϕ + 2πn)(n ɛ Z).But another problem arose, when the mathematicians had to deal withintegrals as∫ z1f(z)dz,z 0where f is a complex-valued function and the limits of integration are complex.Cauchy was the one who came up with the theory of complex integrationin his pamphlet “Mémoire sur les intégrales définies, prises entre des limitesimaginaires” from 1825 (see below). Nevertheless, it seems that the theoryhad already been known to Gauss. In a letter he had written to his friendFriedrich Wilhelm Bessel (1784-1846) in 1811, but not published until 1880,it can be read that Gauss already knew about the “integral theorem”. Atthe end of the letter he says “This is a very beautiful theorem, for which Iwill give a not difficult proof at a suitable opportunity.” But Gauss neverproved the theorem.4 Cauchy and the Integral TheoremCauchy, along with his contemporaries - Gauss, Abel (1802-1829) and Bolzano(1781-1848) - belong to the pioneers of the new insistence on rigor in mathematics.The eighteenth century had been essentially a period of experimentation.When they worried about rigor, as Euler and Lagrange occasionallydid, their arguments were not always convincing [8]. Cauchy, Gauss, Abeland Bolzano are often called the first truly modern mathematicians [1].In this report a biography of Cauchy will be presented and then a summaryof Cauchy’s proof of the integral theorem from his paper “Mémoiresur les intégrales définies, prises entre des limites imaginaires” (1825).4.1 Biography of CauchyAugustin-Louis Cauchy (1789-1857) was born in Paris about five weeks afterthe fall of the Bastille. His mathematical talent was discovered early andhis father introduced him to Laplace (1749-1827) and Lagrange (1736-1813),who both were visitors in the Cauchy family home. Lagrange in particular8
seems to have taken an interest in Cauchy and advised his father “not toallow his son to open a mathematics book nor write a single number before hehas completed his studies in literature” [1]. Otherwise, Lagrange continued,“the boy might be a great mathematician but not capable of writing inhis own native language” [6]. On Lagrange´s advice, Cauchy entered theÉcole Centrale du Panthéon where he spent two years of studying classicallanguages.From 1804 Cauchy attended classes in mathematics and he took theentrance examination for the École Polytechnique in 1805. Two years laterhe graduated from the École Polytechnique and entered the engineeringschool École des Ponts et Chaussées. After that Cauchy spent a few yearsas an engineer in Cherbourg.An academic career was what Cauchy wanted and he applied for a postin the Bureau des Longitudes. He failed to obtain this post and insteadLegendre was appointed. He also lost in 1815, to Binet, a mechanics chairat the École Poytechnique, but then was appointed assistant professor ofanalysis there. In 1816 he won the Grand Prix of the French Academy ofScience for a work on waves. Later he also became professor at Sorbonne.During this time, until 1830, Cauchy´s mathematical activity was incredible.Because of political events Cauchy gave up his professorship and emigratedto Switzerland and Italy for some years. He returned to Paris in 1838 andregained his position at the Academy.Cauchy´s many contributions to the theory of light and to mechanicshave been obscured by the success of his work in analysis [8]. His crowningachievement in analysis was unquestionably his theory of complex functions,a branch of mathematics that commanded his attention from 1814 until hisdeath. Few mathematicians concerned themselves with Cauchy´s theorybefore the late 1840s. Almost all the progress in this area until that timewas due to Cauchy [1].Cauchy´s first significant paper in the theory of functions was “Mémoiresur les intégrales définies”, presented for the Academy of Sciences in 1814.Subsequently his studies on integration expanded in his lessons at the ÉcolePolytechnique, and culminated in his pamphlet “Mémoire sur les intégralesdéfinies, prises entre des limites imaginaires” from 1825. Its object was thedefinite integral whose limits of integration were complex numbers.4.2 The Cauchy Integral TheoremAt the beginning of his pamphlet “Mémoire sur les intégrales définies, prisesentre des limites imaginaires”, Cauchy presents the definition of integrals9
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Page 3 and 4: AbstractIn this report we consider
Page 5 and 6: 1 IntroductionDuring the eighteenth
Page 7 and 8: 3 A short History of Complex Analys
Page 9: ecognition after 1831, when Carl Fr
Page 13 and 14: After that, Cauchy indicates the in
Page 15: 5.1 The Insufficiencies in Cauchy
Page 18 and 19: property of being two telescoping s
Page 20 and 21: and applies Theorem II.It is notabl
Page 22 and 23: that Cauchy didn’t mention this p
Page 24: [7] Smithes, F. (1997). Cauchy and

Malmsten's Proof of the Integral Theorem

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