Taming a Hydra of Singularities

We conclude with four observations: (a) The curves z(s) and z(s, t) were not required to be simple curves. This made the task of approximating them with continuously differentiable curves very easy. (b) The Cauchy integral theorem also holds if the function z(s) describing the curve C is merely piecewise continuously differentiable. For in this case we carry out the integration by parts in (6) over each subinterval on which zs(s) is continuous. It follows from the formula on the right side of (6) that zk s (s) tends to zs(s) uniformly on every subinterval of [0,1] on which zs(s) is continuous. Around points of discontinuity of zs(s), the functions zk s (s) remain uniformly bounded. This is sufficient to conclude that (7) holds as k tends to ∞. (c) Suppose that the closed curve C is homologous to zero in D. Then it can be decomposed as a finite sum of closed curves Cn each of which is homotopic in D to a point. If C is continuously differentiable, each curve Cn is piecewise continuously differentiable, so by what was proved in (b), the integral of f over each curve Cn is zero. But then so is their sum. This shows that the Cauchy integral theorem holds for curves homologous to zero relative to D. (d) The Cauchy integral theorem holds for rectifiable curves, as is easily derived from the result in this note. I don’t know of any application of the Cauchy theorem where we need to go beyond piecewise continuously differentiable curves