Introduction

In previous papers [#!CAoRFI!#,#!CAoRFII!#,#!CAoRFIII!#,#!CAoRFIV!#,#!CAoRFV!#,#!CAoRFVI!#] we have shown that there is a correspondence between, on the one hand, real functions and other real objects on the unit circle, and on the other hand, inner analytic functions within the open unit disk of the complex plane [#!CVchurchill!#]. This correspondence is based on the complex-analytic structure which we introduced in [#!CAoRFI!#]. That complex-analytic structure includes the concept of inner analytic functions, two analytic operations on them, which we named angular differentiation and angular integration, and a scheme for the classification of all the possible singularities of these functions.

This classification scheme separates the singularities as either soft or hard ones, depending on whether or not the limit of the function to the singular point exists. As part of that classification scheme we also introduced gradations of both hardness and softness for the singularities, given by integers degrees. In particular, a hard singularity which becomes soft under a single angular integration of the inner analytic function is a borderline hard one, with degree of hardness zero. In a previous paper [#!CAoRFI!#] we have shown that both soft and borderline hard singularities are integrable ones, while the hard singularities with strictly positive degrees of hardness are non-integrable ones.

Here we will show that one can extend the Cauchy-Goursat theorem, by weakening its hypotheses, so that both soft and borderline hard isolated singularities are allowed on the integration contour. We will prove this first for inner analytic functions, and then generalize the result to arbitrary complex analytic functions, using conformal transformations.

For ease of reference, we include here a one-page synopsis of the complex-analytic structure introduced in [#!CAoRFI!#]. It consists of certain elements within complex analysis [#!CVchurchill!#], as well as of their main properties.