Polar complex numbers in n dimensions

Abstract

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.