The Regularity of Functions on Dual Split Quaternions in Clifford Analysis

Ji Eun Kim,Kwang Ho Shon

doi:10.1155/2014/369430

Abstract

This paper shows some properties of dual split quaternion numbers and expressions of power series in dual split quaternions and provides differential operators in dual split quaternions and a dual split regular function onΩ⊂ℂ2×ℂ2that has a dual split Cauchy-Riemann system in dual split quaternions.

Highlights

Hamilton introduced quaternions, extending complex numbers to higher spatial dimensions in differential geometry
Because of the properties of the eight-unit equality, the addition and subtraction of dual split quaternions are governed by the rules of ordinary algebra
Let Ω be an open set in C2 × C2 and let f = f0 + εf1 = (g0 + g1e2) + ε(g2 + g3e2) be an L1-regular function defined on Ω

Summary

Introduction

Hamilton introduced quaternions, extending complex numbers to higher spatial dimensions in differential geometry (see [1]). = (z0 + z1e2) + ε (z2 + z3e2) = p0 + εp[1], where p0 = z0 + z1e2 and p1 = z2 + z3e2 are split quaternion components, z0 = x0 + x1e1, z1 = x2 + x3e1, z2 = y0 + y1e1, and z3 = y2 + y3e1 are usual complex numbers, and xm, ym ∈ R (m = 0, 1, 2, 3). The multiplication of split quaternionic units with a dual symbol is commutative εer = erε (r = 1, 2, 3).

Results

Conclusion