Clifford Analysis and Related Topics by Paula Cerejeiras & Craig A. Nolder & John Ryan & Carmen Judith Vanegas Espinoza

Author:Paula Cerejeiras & Craig A. Nolder & John Ryan & Carmen Judith Vanegas Espinoza
Language: eng
Format: epub
ISBN: 9783030000493
Publisher: Springer International Publishing

Notions of Regularity for Functions of a Split-Quaternionic Variable

J. A. Emanuello1 and C. A. Nolder1

(1)Department of Mathematics, The Florida State University, 208 Love Building, 1017 Academic Way, Tallahassee, FL 32306-4510, USA

J. A. Emanuello

Email: [email protected]

C. A. Nolder (Corresponding author)

Email: [email protected]

Email: [email protected]

Abstract

The utility and beauty of the theory holomorphic functions of a complex variable leads one to wonder whether analogous function theories exist for other (presumably higher dimensional) algebras. Over the last several decades it has been shown that much of Complex analysis extends to a similar theory for the family of Clifford Algebras . However, there has yet to be a complete description for the general theory over the family of Clifford algebras (for ). In this work, we describe two different approaches from the literature for finding a theory of “holomorphic” functions of a split-quaternionic variable (which is the Clifford algebra ). We show that one approach yields a relatively trivial theory, while the other gives a richer one. In the second instance, we describe a simple subclass of “holomorphic” functions and give two examples of an analogue of the Cauchy-Kowalewski extension in this context.

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