Malmsten's Proof of the Integral Theorem

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that Cauchy didn’t mention this property either in his treatment of definiteintegrals between real limits.Now Malmsten calls∫ zF (z) = f(z)dz (13)z 0and shows that F (z) is synectic in the same domain as f(z). This is analogousto what Cauchy showed for definite integrals between real limits,namely that F (z) is differentiable in the same domain as f(z).The last property to show isHere Malmsten uses expression (13) and putsF ′ (z) = f(z). (14)F (z + δ) − F (z) =∫ z+δzf(z)dzand then uses an earlier corollary (about weighted mean values) to getF (z + δ) − F (z) = δf(z + α) · θ · e pi ,where α is a ’mean quantity’ to 0 and δ (and therefore converges to 0 at thesame time as δ) and θ · e pi converges to 1 at the same time as δ convergesto 0. From this formula he getsF (z + δ) − f(z)lim = F ′ (z) = f(z).δHence, by proving the existence of the integral (1), and then showingthat the properties (10)-(14) holds, Malmsten has now completed his proofof the integral theorem (for integrals where the limits of integration arecomplex numbers) analogous to how Cauchy proved the integral theoremfor integrals between real limits.6 Final RemarksMalmsten left Uppsala at the end of the 1850s to get into politics (seeabove). His successor Herman Daug (1828-1888), whose mathematical researchmostly concerned differential geometry, didn’t have much influence[3]. Instead, in the 1870s, the mathematical education and research in Uppsalacame to be characterized by Göran Dillner (1832-1906), who had agreat interest in Cauchy’s works of analytic functions. But Dillner’s lacking20
knowledge of analysis resulted in a relatively unprofessional way to do research[3]. However, along with Mittag-Leffler, who came under influence ofKarl Weierstrass (1815-1896), the theory of analytic functions got clearedup. In fact, it was in Germany, with Bernhard Riemann (1826-1866) andWeierstrass, the theory of complex functions followed an autonomous pathof development that went far beyond what had been set out by Cauchy [2].Thus, the intuitive apprehension of the analytic functions that had characterizedthe nineteenth century now became rigorous.Malmsten wrote his proof of the integral theorem some years before thebreakthrough of Weierstrass’ strict analysis and therefore his statement ofthe proof doesn’t reach present-day standards. Although, it seems it keptup to the standards of his time. However, the merit of Malmsten is to havesorted out condition (7), even if he didn’t use it in Green’s formula. Theproof also shows Malmsten’s ability to treat inequalities as well as Riemannsums. Furthermore, it is said that Mittag-Leffler, who at the beginningof his career had been one of Malmsten’s students, was complaining thatMalmsten’s, as well as his own, proof didn’t end up in “Enzyklopädie derMathematischen Wissenschaften” [3]. Maybe this was due to the fact thatMalmsten’s successor Daug didn’t keep up the research in the theory ofanalytic functions.References[1] Belhoste, B. (1991). Augustin-Louis Cauchy. A Biography. New York:Springer-Verlag.[2] Bottazzini, U. (1986). The Higher Calculus: A History of Real and ComplexAnalysis from Euler to Weierstrass. New York: Springer-Verlag.[3] Gårding, L. (1994). Matematik och matematiker. Matematiken i Sverigeföre 1950. Lund: Lund University Press.[4] Kolmogorov, A.N., Yushkevich, A.P. (1996). Mathematics of the 19thCentury. Basel: Birkhäuser Verlag.[5] Malmsten, C.J. (1865). Om definita integraler mellan imaginära gränsor.Kungliga Svenska Vetenskaps-akademiens Handlingar. Bandet 6. N:o 3.[6] Sjöberg, B. (1998). Från Euklides till Hilbert: historien om matematikensutveckling under tvåtusen år. Åbo: Åbo Akademis förlag.21
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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
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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 24: [7] Smithes, F. (1997). Cauchy and

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