2000/08/16 by Silviu Olariu, Olariu, Silviu
Mathematics · #30G35 (Primary) 32A45 #33E20 #46F15 #58J15 (Secondary) #Algebraic and Geometric Analysis #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/0008123
27 pages, 4 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
Two distinct systems of commutative complex numbers in 6 dimensions of the polar and planar types of the form u=x0+h1x1+h2x2+h3x3+h4x4+h5x5 are described in this work, where the variables x0, x1, x2, x3, x4, x5 are real numbers. The polar 6-complex numbers introduced in this paper can be specified by the modulus d, the amplitude ρ, and the polar angles θ+, θ-, the planar angle ψ1, and the azimuthal angles ϕ1, ϕ2. The planar 6-complex numbers introduced in this paper can be specified by the modulus d, the amplitude ρ, the planar angles ψ1, ψ2, and the azimuthal angles ϕ1, ϕ2, ϕ3. Exponential and trigonometric forms are given for the 6-complex numbers. The 6-complex functions defined by series of powers are analytic, and the partial derivatives of the components of the 6-complex functions are closely related. The integrals of polar 6-complex functions are independent of path in regions where the functions are regular. The fact that the exponential form of ther 6-complex numbers depends on cyclic variables leads to the concept of pole and residue for integrals on closed paths. The polynomials of polar 6-complex variables can be written as products of linear or quadratic factors, the polynomials of planar 6-complex variables can always be written as products of linear factors, although the factorization is not unique.