Polar complex numbers in n dimensions
| dc.creator | Olariu, Silviu | |
| dc.date | 2000-08-16 | |
| dc.date.accessioned | 2026-07-07T04:36:50Z | |
| dc.date.available | 2026-07-07T04:36:50Z | |
| dc.description | Polar commutative n-complex numbers of the form u=x_0+h_1x_1+h_2x_2+...+h_{n-1}x_{n-1} are introduced in n dimensions, the variables x_0,...,x_{n-1} being real numbers. The polar n-complex number can be represented, in an even number of dimensions, by the modulus d, by the amplitude ρ, by 2 polar angles θ_+,θ_-, by n/2-2 planar angles ψ_{k-1}, and by n/2-1 azimuthal angles ϕ_k. In an odd number of dimensions, the polar n-complex number can be represented by d, ρ, by 1 polar angle θ_+, by (n-3)/2 planar angles ψ_{k-1}, and by (n-1)/2 azimuthal angles ϕ_k. The exponential function of a polar n-complex number can be expanded in terms of the polar n-dimensional cosexponential functions g_{nk}(y), k=0,1,...,n-1. Expressions are given for these cosexponential functions. The polar n-complex numbers can be written in exponential and trigonometric forms with the aid of the modulus, amplitude and the angular variables. The polar n-complex functions defined by series of powers are analytic, and the partial derivatives of the components of the polar n-complex functions are closely related. The integrals of polar n-complex functions are independent of path in regions where the functions are regular. The fact that the exponential form of a polar n-complex numbers depends on the cyclic variables ϕ_k leads to the concept of pole and residue for integrals on closed paths. The polynomials of polar n-complex variables can be written as products of linear or quadratic factors, although the factorization may not be unique. | |
| dc.description | 38 pages, 3 figures | |
| dc.identifier | https://arxiv.org/abs/math/0008124 | |
| dc.identifier | http://arxiv.org/abs/math/0008124 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/59745 | |
| dc.subject | Complex Variables | |
| dc.subject | 30G35 (Primary) 32A45, 33E20, 46F15, 58J15 (Secondary) | |
| dc.title | Polar complex numbers in n dimensions | |
| dc.type | text |