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Improving Cauchy's Theorem in Constructive Analysis

Douglas S. Bridges

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.

Improving Cauchy's Theorem in Constructive 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.

Paper Structure

This paper contains 2 sections, 5 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Extending Cauchy's theorem

Key Result

Theorem 1

Let the piecewise differentiable closed paths $\gamma_{0}$ and $\gamma_{1}\ $have common parameter interval and be homotopic in the open set $U\subset\mathbb{C}$. Then $\int_{\gamma_{0}}f=\int_{\gamma_{1}}f$ for each analytic function $f$ on $U$.

Theorems & Definitions (5)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Corollary 4
  • Corollary 5