Extensions of the Cauchy–Goursat Integral Theorem

It is natural to conjecture that if a function f is continuous on the closed region determined by a rectifiable 1-cycle Γ and complex-differentiable on the open region then ∫Γf=0. The main result is an extension of the classical Cauchy–Goursat Theorem: the equality conjectured holds (with no boundary condition on f′) under the additional hypothesis that the winding numbers of Γ define an Lp function and f satisfies a matching Hölder continuity condition near the image of Γ. (In particular, continuity suffices if p=∞.) The proof uses approximations of a rectifiable path by piecewise linear paths.