On the Quadratic Cone of ${\mathbb{R}}_3.$

Abstract

In this paper we study the following type of functions $f:{\mathcal{Q}}_{{\mathbb{R}}_3}\to {\mathbb{R}}_3$, where ${\mathcal{Q}}_{{\mathbb{R}}_3}$ is the quadratic cone of the algebra ${\mathbb{R}}_3$. From the fact that it is possible to write the algebra ${\mathbb{R}}_3$ as a direct sum of quaternions, we get the observation that it is possible to find a clever representation for ${\mathcal{Q}}_{{\mathbb{R}}_3}$. By using this result a slice regular theory was introduced and a Cauchy formula is discussed. Moreover, a detailed study of the zeros is performed. Finally, we find a formula for the determinant of a matrix with entries in ${\mathcal{Q}}_{{\mathbb{R}}_3}$.

References in corpus (7)

• Slice regular functions on real alternative algebras
• Poles of regular quaternionic functions
• The Schwarz-Pick lemma for slice regular functions
• Singularities of slice regular functions over real alternative *-algebras
• Power and spherical series over real alternative *-algebras
• Slice-quaternionic Hopf surfaces
• On a quaternionic Picard theorem