Components exponential scator holomorphic function

Abstract

The components exponential scator function, succinctly labeled "cexp," is a function of scator variable in 1 + n dimensional elliptic scator algebra. This function is introduced here, and its elementary properties are studied. In 1 + 1 dimensions, the cexp function becomes the usual complex exponential function of complex variable. The cexp function is shown to be scator holomorphic in the entire scator set according to the differential quotient criterion. The scator derivative of the cexp function is the cexp function itself. The relationship between the Cartesian and polar forms of the cexp function can be seen as a higher-dimensional extension of Euler formula. The mappings of grids in 1 + 2 dimensional space exhibit ellipses and Lissajous-like figures in addition to circles and radial lines. A cusphere surface is generated for the isometric condition in the function's image. An interesting application of the cexp function as propagators regarding the quantum measurement problem is outlined. KEYWORDS holomorphic functions, hypercomplex variables 2 SCATOR ALGEBRA PREAMBLE Elliptic scator algebra is a 1+n finite dimensional algebra. In the additive form, the elements of this algebra are represented by o = (z 0 ; z 1 , … , z n ) = z 0 + n ∑ =1 zě ,Math Meth Appl Sci. wileyonlinelibrary.com/journal/mma

Powers of Elliptic Scator Numbers

“…The o cexp function is scator holomorphic in the entire R 1+n domain according to the differential quotient criterion [11]. In the additive representation, the…”

“…The product can be written with multiplicative variables in the additive representation from ( 5) and (11),…”

“…The multiplicative (11) and additive (9a)-(9c) definitions of the product are equivalent. In the early papers, this equivalence led us to a wrong statement regarding the product associativity in the additive representation, that has now been corrected [8].…”

“…Therefore, for n > 1, the product of unit magnitude scators will remain unitary if the scalar component of the products and its scator conjugates do not vanish. In particular, the powers of associative unit scators lie on the isometric scator surface, called a cusphere [11]. The unit cusphere is an n dimensional surface embedded in a 1 + n dimensional space that satisfies the condition…”

“…The relationship between the two scator representations is established by a generalized Euler's type equation. The components exponential function is a scator function of scator argument that extends the notion of the complex exponential to higher dimensions [11]. This function, labeled 'cexp', is scator holomorphic and its derivative is the function itself, thus it satisfies the differential equation f In this paper, the essentials of elliptic scator algebra are presented in section 2.…”