Complex Analysis of Real Functions VII: A Simple Extension of the Cauchy-Goursat Theorem

TL;DR

This paper extends the Cauchy-Goursat theorem to include cases where the complex function has isolated integrable singularities on the contour, broadening its applicability within complex analysis.

Contribution

It provides a simple generalization of the Cauchy-Goursat theorem for functions with isolated integrable singularities on the contour.

Findings

The theorem holds for inner analytic functions within the unit disk.

The generalization applies to all analytic functions with isolated integrable singularities.

It broadens the scope of contour integration in complex analysis.

Abstract

In the context of the complex-analytic structure within the open unit disk, that was established in a previous paper, here we establish a simple generalization of the Cauchy-Goursat theorem of complex analytic functions. We do this first for the case of inner analytic functions, and then generalize the result to all analytic functions. We thus show that the Cauchy-Goursat theorem holds even if the complex function has isolated singularities located on the integration contour, so long as these are all integrable ones.