Improving Cauchy's Theorem in Constructive Analysis

TL;DR

This paper revises the constructive proof of Cauchy's integral theorem in Bishop-style mathematics, using standard definitions of homotopy and simple connectedness, thereby broadening the theorem's applicability.

Contribution

It provides a constructive proof of Cauchy's theorem with less restrictive notions, aligning Bishop's results with more conventional definitions.

Findings

Cauchy's integral theorem holds under standard notions in constructive analysis.

Bishop's theorems are valid with less restrictive definitions.

The proof broadens the applicability of constructive complex analysis.

Abstract

In his constructive development of complex analysis, Errett Bishop used restrictive notions of homotopy and simple connectedness. Working in Bishop-style constructive mathematics, we prove Cauchy's integral theorem using the standard notions of such properties. In consequence, Bishop's theorems in Chapters 5 of [1, 2] hold under our more normal, less restrictive, definitions.