Malmsten's Proof of the Integral Theorem

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Contents1 Introduction 32 Biography of Malmsten 33 A short History of Complex Analysis 53.1 Development of the Eighteenth Century . . . . . . . . . . . . 53.2 The Geometric Representation of a Complex Number . . . . 63.3 Complex Integration . . . . . . . . . . . . . . . . . . . . . . . 74 Cauchy and the Integral Theorem 84.1 Biography of Cauchy . . . . . . . . . . . . . . . . . . . . . . . 84.2 The Cauchy Integral Theorem . . . . . . . . . . . . . . . . . . 95 Malmsten´s Article: “Om definita integraler mellan imaginäragränsor” 125.1 The Insufficiencies in Cauchy’s Proof According to Malmsten 135.2 How Malmsten Proves the Existence of the Integrals . . . . . 145.3 The Various Properties of the Integral . . . . . . . . . . . . . 196 Final Remarks 202
1 IntroductionDuring the eighteenth century the mathematicians in Uppsala were mainlyastronomers, physicists or theologians. In fact, it wasn’t until the 1820sthat Uppsala, as well as Lund, got their first pure mathematicians. Maybethis was a natural ongoing process toward specialization. It could also havebeen due to the fact that the rumors of great works of Euler, d’Alembert,Lagrange and Laplace now had reached the universities of Scandinavia [3]. InUppsala, at the time, a new generation of talented mathematicians appeared.Among the most prominent were Adolf Fredrik Svanberg (1806-1857), CarlJohan Malmsten (1814-1886) and Emanuel Gabriel Björling (1808-1872).In this report we discuss the article “Om definita integraler mellan imaginäragränsor”, which was written by C. J. Malmsten in 1865. In his work,Malmsten gives a proof of Cauchy’s integral theorem, i.e. that the integralis independent of the path of integration (where the limits of integrationare complex numbers). The aims of this report is to try to catch the techniquesthat Malmsten made use of in his proof and to get a glimpse of themathematical concepts at this time, especially those which weren’t fully investigatedand thereby gave rise to some problems for the mathematicians.The intention is also, to some extent, to become acquainted with the mathematicalarena in Sweden at this time.To give a background to Malmsten’s article and to see how far the developmentof complex analysis had reached at this time, we begin with ashort history of this field. Subsequently, a summary of Cauchy’s proof ofthe integral theorem will be presented, followed by a biography of Cauchy,before the exposition of Malmsten’s article. However, the report begins witha biography of Malmsten.2 Biography of MalmstenCarl Johan Malmsten (1814-1886) grew up in Uddetorp, not far from Skara.Shortly after the death of his father (1827) Malmsten, who at the time wasin ’Skara skola’, obtained a favourable position as a tutor. This post made itpossible for Malmsten to study without interruptions and he was able to keepthe post during his upper secondary studies as well as his university studies.In 1833 Malmsten began to study mathematics at Uppsala university wherehe 1839 received a doctoral degree.In competition with E. G. Björling (1808-1872) Malmsten became professorof mathematics in 1841 and thereby succeeded his own teacher Jöns3
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Page 24: [7] Smithes, F. (1997). Cauchy and

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