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Advances in Applied Clifford Algebras

, Volume 8, Issue 2, pp 323–340 | Cite as

Holomorphic Cliffordian functions

  • Guy Laville
  • Ivan Ramadanoff
Papers

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.

Keywords

Holomorphic Function Clifford Algebra Laurent Series Polynomial Solution Monogenic Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser-Verlag AG 1998

Authors and Affiliations

  1. 1.UPRES-A 6081 Département de MathématiquesUniversité de CaenCaen CedexFrance

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