Malmsten's Proof of the Integral Theorem

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