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Clifford Analysis

In: Lectures on Clifford (Geometric) Algebras and Applications

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  • John Ryan

    (University of Arkansas, Department of Mathematics)

Abstract

We introduce the basic concepts of Clifford analysis. This analysis started many years ago as an attempt to generalize one variable complex analysis to higher dimensions. Most of the basic analysis was initially developed over the quaternions which are a division algebra. However, it was soon realized that virtually all of this analysis extends to all dimensions using Clifford algebras. Here we introduce a generalized Cauchy-Riemann operator, often called a Dirac operator, and the analogues of holomorphic functions. These functions are called Clifford holo-morphic functions or monogenic functions. We give a generalization of Cauchy’s theorem and Cauchy’s integral formula. Using Cauchy’s theorem, we can establish the Möbius invariance of monogenic functions. We will also introduce the Plemelj formulas and operators, and Hardy spaces.

Suggested Citation

  • John Ryan, 2004. "Clifford Analysis," Springer Books, in: Rafal Abłamowicz & Garret Sobczyk (ed.), Lectures on Clifford (Geometric) Algebras and Applications, chapter 3, pages 53-89, Springer.
    • Handle: RePEc:spr:sprchp:978-0-8176-8190-6_3
    DOI: 10.1007/978-0-8176-8190-6_3
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