2000/08/16 by Silviu Olariu, Olariu, Silviu
Mathematics · Physics and Astronomy · #30G35 (Primary) 32A45 #33E20 #46F15 #58J15 (Secondary) #Advanced Mathematical Theories and Applications #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/0008122
18 pages, 2 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
A system of commutative complex numbers in 5 dimensions of the form u=x0+h1x1+h2x2+h3x3+h4x4 is described in this paper, the variables x0, x1, x2, x3, x4 being real numbers. The operations of addition and multiplication of the 5-complex numbers introduced in this work have a geometric interpretation based on the the modulus d, the amplitude ρ, the polar angle θ+, the planar angle ψ1, and the azimuthal angles ϕ1,ϕ2. The exponential function of a 5-complex number can be expanded in terms of polar 5-dimensional cosexponential functions g5k(y), k=0,1,2,3,4, and the expressions of these functions are obtained from the properties of the exponential function of a 5-complex variable. Exponential and trigonometric forms are obtained for the 5-complex numbers, which depend on the modulus, the amplitude and the angular variables. The 5-complex functions defined by series of powers are analytic, and the partial derivatives of the components of the 5-complex functions are closely related. The integrals of 5-complex functions are independent of path in regions where the functions are regular. The fact that the exponential form of the 5-complex numbers depends on the cyclic variables ϕ1, ϕ2 leads to the concept of pole and residue for integrals on closed paths. The polynomials of 5-complex variables can be written as products of linear or quadratic factors.