JP Journal of Geometry and Topology
Volume 9, Issue 3, Pages 263 - 271
(November 2009)
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A CAUCHY-LIKE THEOREM FOR HYPERCOMPLEX FUNCTIONS
M. F. Borges, J. A. Marão and R. C. Barreiro
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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. |
Communicated by Yasuo Matsushita |
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