, Volume 8, Issue 2, pp 323-340

Holomorphic Cliffordian functions

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Abstract

The aim of this paper is to put the foundations of a new theory of functions, called holomorphic Cliffordian, which should play an essential role in the generalization of holomorphic functions to higher dimensions. Let ℝ0,2m+1 be the Clifford algebra of ℝ2m+1 with a quadratic form of negative signature, \(D = \sum\limits_{j = 0}^{2m + 1} {e_j {\partial \over {\partial x_j }}} \) be the usual operator for monogenic functions and Δ the ordinary Laplacian. The holomorphic Cliffordian functions are functionsf: ℝ2m+2 → ℝ0,2m+1, which are solutions ofDδ m f = 0.

Here, we will study polynomial and singular solutions of this equation, we will obtain integral representation formulas and deduce the analogous of the Taylor and Laurent expansions for holomorphic Cliffordian functions.

In a following paper, we will put the foundations of the Cliffordian elliptic function theory.