Analysis over Cayley-Dickson numbers and its applications. Hypercomplex holomorphic functions, meromorphic functions, partial differential equations, operational calculus
type: Item
instance of: scholarly article
zbMATH Open document ID: 1257.30001
author: Sergey V. Ludkovsky
publication date: 2012-10-05T00:00:00Z
Mathematics Subject Classification ID: 30-02
Mathematics Subject Classification ID: 30G35
Mathematics Subject Classification ID: 17A35
zbMATH DE Number: 6090399
zbMATH Keywords: Cayley-Dickson numbers
zbMATH Keywords: analysis
zbMATH Keywords: complex numbers
zbMATH Keywords: quaternions
zbMATH Keywords: octonions
zbMATH Keywords: sedenions
zbMATH Keywords: partial differential equations
MaRDI profile type: Publication
title: Analysis over Cayley-Dickson numbers and its applications. Hypercomplex holomorphic functions, meromorphic functions, partial differential equations, operational calculus
review text: The book deals with function theory over so-called Cayley-Dickson numbers, which are non-associative generalizations of real quaternions. Cayley-Dickson algebras contain complex numbers, quaternions, octonions, sedenions (sedonions) and so on. The octonion algebra is not longer associative, but at least alternative. Sedenions and higher dimensional algebras in this row do not longer have to be alternative. It should be mentioned that all Clifford algebras are associative, i.e. they can be represented by matrices over the fields of real numbers, complex numbers or quaternions. This is in Cayley-Dickson algebras not longer the case. Nevertheless, Cayley-Dickson algebras have a rich structure, and hence in some way this also applies to the analysis over them. In some sense, a similar theory to complex analysis can be successfully developed. This analysis is called in this book super-analysis. Quantum mechanics, quantum field theory and super-gravitational theories motivate to study such a general topic.NEWLINENEWLINEChapter 1 introduces the notion of super-differentiability over domains in Cayley-Dickson algebras, a corresponding function theory is developed. There is a natural connection between super-differentiability and Dirac spinors in quantum field theory.NEWLINENEWLINEThe second chapter is devoted to the theory of meromorphic functions with Cayley-Dickson variables. Elements of a residue theory are presented. Residues are alway operators here. Algorithms for their computations are described and demonstrated by examples.NEWLINENEWLINE In the third chapter some types of partial differential equations over Cayley-Dickson algebras are considered; a method adapted from ordinary differential equations is used. Examples are taken over to octonions. Also, some non-linear differential equations can be investigated.NEWLINENEWLINEIn Chapter 4 a non-commutative integration method for the solution of partial differential equations in mathematical physics, optics and quantum …
description: scientific article; zbMATH DE number 6090399