2000/08/16 by Silviu Olariu, Olariu, Silviu
Mathematics · Physics and Astronomy · #30G35 (Primary) 32A45 #33E20 #46F15 #58J15 (Secondary) #Advanced Mathematical Theories and Applications #Complex Variables (math.CV) #FOS: Mathematics #Mathematics and Applications #math.CV #msc:30G35 #msc:32A45 #msc:33E20 #msc:46F15 #msc:58J15
paper · pdf · doi:10.48550/arxiv.math/0008125
32 pages, 3 figures
arxiv created 2000/08/16 · openalex publication_date 2000/08/16 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Planar commutative n-complex numbers of the form u=x0+h1x1+h2x2+...+hn-1xn-1 are introduced in an even number n of dimensions, the variables x0,...,xn-1 being real numbers. The planar n-complex numbers can be described by the modulus d, by the amplitude ρ, by n/2 azimuthal angles ϕk, and by n/2-1 planar angles ψk-1. The exponential function of a planar n-complex number can be expanded in terms of the planar n-dimensional cosexponential functions fnk, k=0,1,...,n-1, and expressions are given for fnk. Exponential and trigonometric forms are obtained for the planar n-complex numbers. The planar n-complex functions defined by series of powers are analytic, and the partial derivatives of the components of the planar n-complex functions are closely related. The integrals of planar n-complex functions are independent of path in regions where the functions are regular. The fact that the exponential form of the planar n-complex numbers depends on the cyclic variables ϕk leads to the concept of pole and residue for integrals on closed paths. The polynomials of planar n-complex variables can always be written as products of linear factors, although the factorization may not be unique.