Malmsten's Proof of the Integral Theorem

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to arrive at the system of equations∂u∂x = ∂v∂y ,∂u∂y = − ∂v∂x(today known as the Cauchy-Riemann equations). d´Alembert, and afterhim Euler, obtained pairs of solutions of this system as the real and imaginaryparts of (what we today would call) an analytic function f(z) = u + ivof a complex variable.In Eulers work on integral calculus he established the equality∫∫∫f(z)dz = (udx − vdy) + i (vdx + udy),where f(z) = u + iv. Only the concept of integrals of functions of a complexvariable computed along some curve was lacking to here see the Cauchyintegral theorem that the integral is independent of the path of integration.In the work of Euler, as well as d´Alembert, complex numbers enter thediscussion usually as pairs of real numbers having a certain specific meaning,for example as pairs formed from the abscissa and ordinate of a pointin a plane, the projections of the velocity of a particle of fluid on the coordinateaxes, or pairs of functions. Thus the various particular interpretationsof complex numbers (in particular, geometrical interpretations) were incorporatedinto mathematics (in some sense) starting in the mid-eighteenthcentury [4]. However a visual representation of the complex variable as avector or a point moving over the plane was still missing, even though Eulerwas quite close to such a concept in his paper “On the representation of aspherical surface on a plane” (1777) [4].3.2 The Geometric Representation of a Complex NumberThe first explicit and systematic geometric representation of complex numbersas directed line segments and the corresponding interpretation of operationson them occur in a 1799 paper by the Danish surveyor Caspar Wessel(1745-1818). But Wessel´s work went completely unnoticed until it was rediscovereda hundred years later and republished in a French translation[2]. Other roughly contemporary publications had also gone unnoticed, forexample, by abbot Adrien Buée (1748-1826) in 1806, and that of the SwissJean-Robert Argand (1768-1822), who published an anonymous pamphleton the geometric interpretation of complex numbers at his own expense [2].However, in the first quarter of the nineteenth century many mathematicianswere very close to the geometric representation of complex numbers.The concept of complex numbers as points of a plane received general6
ecognition after 1831, when Carl Friedrich Gauss (1777-1855) published thepaper “Theoria residuorum biquadraticorum”. This paper included a justificationof the complex numbers and a geometric interpretation of them.The term “complex numbers” itself is due to Gauss [4].Cauchy had a completely different approach explaining the foundationsof the theory of complex numbers. In his Analyse algébrique (1821) complexquantities were introduced in a formal manner. He classifies most “imaginaryexpressions” (i.e., complex numbers) and “imaginary equations” (i.e.,equalities containing complex numbers) as symbolic expressions. His viewof the meaning of these expressions when taken literally is...imprecise or devoid of meaning, but from which one candeduce precise results by modifying and altering according to definiterules, either the equations themselves or the symbolic containedin them.It wasn´t until 1847 in the paper “Mémoire sur les quantités géométriques”(1847) that Cauchy replaced the imaginary expressions by the geometricalinterpretation. Although, it is curious that when Cauchy had proved thetheorem that the integral is independent of the path of integration (seebelow) he still emphasized that imaginary numbers “do not refer to anythingand have no meaning. The sign √ −1 is in a way only a device, a tool forcomputation...” [4].3.3 Complex IntegrationAt the beginning of the eighteenth century mathematicians began to operatewith complex-valued functions. These were always separated into a real andan imaginary part, so they could be treated like real-valued functions. Asan example of an early use of complex-valued functions Leibniz as well asJohann Bernoulli (1667-1748) evaluated the real integral∫dxax 2 + bx + cby splitting up the integrand into two partial fractions and then doing theintegration. The fact that if the roots to the equationax 2 + bx + c = 0are complex, the denominator will also be complex, didn´t stop them toperform the integration in the same way as if the denominators were real7
Page 1 and 2: Malmsten's Proof of the Integral Th
Page 3 and 4: AbstractIn this report we consider
Page 5 and 6: 1 IntroductionDuring the eighteenth
Page 7: 3 A short History of Complex Analys
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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