Malmsten's Proof of the Integral Theorem

After that, Cauchy indicates the integral (1) by A+iB and substitutes for xand y the functions of t that were defined above. Thus, he gets the integralof the formwhich, by settingcan be writtenA + iB =∫ TA + iB =t 0[ϕ ′ (t) + iψ ′ (t)]f[ϕ(t) + iψ(t)]dtϕ ′ (t) = x ′ , ψ ′ (t) = y ′ ,∫ Tt 0(x ′ + iy ′ )f(x + iy)dt.At this stage Cauchy states his integral theorem as follows:“We now suppose that the function f(x + iy) remains finiteand continuous whenever x remains between the limits x 0 andX. In this particular case we can easily prove that the value ofthe integral (1), that is to say, the imaginary expression A + iB,is independent of the nature of the functions x = ϕ(t) and y =ψ(t).”Thus, we can see that Cauchy has reduced an integral between complexlimits to an integral of a complex-valued function between real limits. Theusual formulation can easily be obtained if we observe that the functionsx = ϕ(t) and y = ψ(t) can be thought of as the parametrical equationsof a curve in the complex plane joining the points (x 0 + iy 0 ) and X + iY.But, as mentioned above, Cauchy didn’t have a geometrical interpretationof complex numbers at this time.The hypotheses for f(z) - that it is finite and continuous - is howeverinsufficient. In fact, Cauchy tacitly assumes and implicitly uses both theexistence and the continuity of f ′ (z). This is due to Cauchy’s conviction thata continuous function is always differentiable and that its derivative can bediscontinuous only at points where the function itself is discontinuous [4].Later Cauchy considers this proof to be not rigorous enough. Instead heuses the techniques of calculus of variations. Given a curveCauchy considers a second curve(x(t), y(t)) (3)(x(t) + u(t), y(t) + v(t)), (4)11