A CAUCHY-LIKE THEOREM FOR HYPERCOMPLEX FUNCTIONS

M. F. Borges, J. A. Marão and R. C. Barreiro

Received June 2, 2009; Revised October 6, 2009

Abstract

The classical theory of functions of a complex variable presents important contributions for the development of mathematical analysis and geometry [1, 4, 5, 11]: (i) basic properties of holomorphic functions may deduced from the well-known Cauchy Integral Formula; (ii) conformal mappings, Riemann’s mappings theorem, analytic function on a Riemann surface, and ultimately the Riemann-Roch and Abel Theorems [11] are all studied from the basics of complex theory. Extensions of complex numbers, such as quaternions, exhibit properties and results such as a set of Cauchy-Riemann like relations [13], conformal and quasiconformal transformations [14, 15] and generalized Cauchy and Morera theorems. In this paper, with the purpose of giving a simple geometrical foundation for hypercomplex, we revisit the Cauchy theorem and obtain a Cauchy-like integral hypercomplex formula based on recent results on hyper regularity and hyper periodicity of the hypercomplex exponential function [2, 3].

Keywords and phrases: Cauchy integral, hypercomplex, quaternions.