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- W4375959191 abstract "In this paper, we give a new characterization of the split-complex numbers as a vector space $LC_2= {xI+yE : x,y in mathbb{R},, E^2=I }$ 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 $LC_2 $. We study some properties of the elements of $LC_2$, the trace, the determinant, invertibility conditions, and others. We study some elementary functions defined on subsets of $LC_2$ as compared and contrasted with the usual complex functions. We study properties like continuity, differentiability and define the holomorphic condition of $LC_2$ functions in a different sense than complex functions. We establish the line integrals of the vector-valued functions in $LC_2$ and compare them against the well known results for complex functions of a complex variable." @default.
- W4375959191 created "2023-05-10" @default.
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- W4375959191 date "2023-05-08" @default.
- W4375959191 modified "2023-09-26" @default.
- W4375959191 title "New characterizations of the ring of the split-complex numbers and the field $ mathbb{C} $ of complex numbers and their comparative analyses" @default.
- W4375959191 doi "https://doi.org/10.48550/arxiv.2305.04586" @default.
- W4375959191 hasPublicationYear "2023" @default.
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