Malmsten's Proof of the Integral Theorem

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Svanberg (1771-1851) [3]. In the same year Malmsten was one of the initiatorsof Frey, a journal of art and science, which was published in ten volumes[9].During the 1840s Malmsten wrote many remarkable articles and becameafter Samuel Klingenstierna (1698-1765) the first Swedish mathematicianwho not only followed the development of mathematics of his time, butalso contributed to it [3]. He had a gift for languages and his lectures wereappreciated also by pure humanists. He was a friend of Geijer and took partin the social life of ’Malla Silfverstolpes salong’. One can say that Malmstencontributed to the fact that mathematics became a rather popular topicduring the 1840s [9].In his own mathematical research, Malmsten took an active part in analysis,algebra and probability theory. In a summary of Malmsten’s mathematicalresearch, Gösta Mittag Leffler (1846-1927) emphasizes his calculationof the remainder term in Euler’s “summation” formula as well as hissolutions of certain differential equations [9].Malmsten had a genuine interest in probability theory, which mainlyresulted in his essays and popular lectures about the insurance business. Itis said that one of the motives was his sympathy for the misfortunate [9]. In1855 Malmsten was one of the founders of the insurance company ’Skandia’,where he was a director for many years.In 1858 Malmsten became a member of a committee, whose task was toinvestigate the pensions system. In 1859 he was appointed minister withoutportfolio and in 1866 county governor of ’Skaraborgs län’. In the course oftime Malmsten did regret that he had given up science for politics. One ofthe reasons that he didn’t return to his professorship was consideration forhis substitute Daug, whose financial position was poor [9].In public life Malmsten became famous for a brilliant speech concerningthe freedom for the press at the fiftieth anniversary of the constitution andfor his excellent dinners at the county governor’s house in Mariestad [3].The article “Om definita integraler mellan imaginära gränsor” was writtenafter Malmsten’s years as professor (1865). A few years later the Cauchyintegral theorem, as well as the whole theory of analytic functions, wouldprovide a lot of problems for the mathematicians in Uppsala. It also gaverise to their own interpretations of the theorem.4
3 A short History of Complex AnalysisIt took quite a while before the mathematicians began to accept the complexnumbers as independent mathematical objects. One of the reasons mighthave been the relatively late entrance of negative numbers on the mathematicalscene [6]. In fact, mathematicians didn´t accept negative numbersas roots of equations until the sixteenth century.As soon as mathematicians were able to solve the third- and fourthdegreeequations, another problem arose. In Cardano´s (1501-1576) formula(for solving third-degree equations) square roots of negative numberssometimes appeared, which confused mathematicians of the time. RafaelBombelli (1526-1573) was one of the first who tried to operate with thesenumbers. But his methods were not very successful and he was also verysceptical of these numbers.During the seventeenth century the development of the complex numbersdidn´t make much progress and mathematicians such as Isaac Newton(1642-1727) and Wilhelm Leibniz (1646-1716) displayed a sceptical attitudeto these numbers. Leibniz considered them to lie somewhere in between existenceand non-existence and compared them to The holy spirit [6]. Althoughhe made some calculations with complex numbers, he didn’t understand themeaning of them.3.1 Development of the Eighteenth CenturyDuring the eighteenth century the development of complex analysis accelerated.An entirely new theme of this time was the use of complex values ofan independent variable. The leading man of this work was Leonhard Euler(1707-1783).In the very first pages of the first volume of his Introductio in analysininfinitorum (1748), Euler asserted that a variable can assume any valuewhatsoever, including an imaginary one. For example, in order to clarifythat “a function of a variable quantity is itself a variable quantity,” heasserted that there is no value which the function is not capable of assuming,“since a variable quantity also includes imaginary values” [2]. In the workof Euler the theory of elementary functions of a complex variable received afull development and was perfected by the middle of the eighteenth century[4].Euler established that the equation ∂u/∂x = ∂v/∂y is the condition underwhich v dx+u dy is the exact differential of some function. In connectionwith problems in fluid mechanics Jean d´Alembert (1717-1783) was the first5
Page 1 and 2: Malmsten's Proof of the Integral Th
Page 3 and 4: AbstractIn this report we consider
Page 5: 1 IntroductionDuring the eighteenth
Page 9 and 10: ecognition after 1831, when Carl Fr
Page 11 and 12: seems to have taken an interest in
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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