New characterizations of the ring of the split-complex numbers and the field $\mathbb{C}$ of complex numbers and their comparative analyses

Abstract

In this paper, we give a new characterization of the split-complex numbers as a vector space $L{C}_2=\{xI+yE:x,y\in \mathbb{R},E^2=I\}$ of operators, where $I$ is the identity operator and $E$ is the unit shift operator that are operating on the space ${\mathbb{P}}_2$ of all real-valued $2$-periodic functions. We also characterize the field of the complex number $\mathbb{C}=\{x+yi:x,y\in \mathbb{R},i^2=-1\}$ as the space of linear operators of the form $\{xI+yE,E^2=-I\}$, where $I$ is the identity operator and $E$ the unit shift operator that are regarded as operating on the vector space ${\mathbb{AP}}_2$ of all real-valued $2$-antiperiodic functions. In an analogy to the polar form of complex numbers, we form the hyperbolic form of some subset $\mathcal{H}$ of the elements of $L{C}_2$. We study some properties of the elements of $L{C}_2$, the trace, the determinant, invertibility conditions, and others. We study some elementary functions defined on subsets of $L{C}_2$ as compared and contrasted with the usual complex functions. We study properties like continuity, differentiability and define the holomorphic condition of $L{C}_2$ functions in a different sense than complex functions. We establish the line integrals of the vector-valued functions in $L{C}_2$ and compare them against the well known results for complex functions of a complex variable.

26 pages, 2 figures, one table