On the Cauchy Integral Theorem and Polish spaces

TL;DR

This paper extends the Cauchy Integral Theorem to functions that are continuous in an open set and analytic outside a Polish space with specific characteristics, showing the integral along triangle boundaries vanishes.

Contribution

It introduces a new class of singularities characterized by Polish spaces with specific properties, generalizing classical Cauchy theorem conditions.

Findings

The integral of such functions along triangle boundaries is zero.

Polish space characteristics influence the analyticity and integral properties.

The result broadens the scope of the Cauchy Integral Theorem.

Abstract

We prove that if a function $f$ is continuous in an open subset $U\subset\mathbb{C}$ and analytic in $U\setminus X$, where $X\subset U$ is a Polish space having characteristic system $(i,n)$, such that $i\in\{0,1\}$ and $n\in\mathbb{N}$, then the complex integral line of $f$ along the boundary of any triangle in $U$ vanishes.