Planar complex numbers in even n dimensions

Abstract

Planar 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 an even number n of dimensions, the variables x_0,...,x_{n-1} being real numbers. The planar n-complex numbers can be described by the modulus d, by the amplitude \rho, by n/2 azimuthal angles \phi_k, and by n/2-1 planar angles \psi_{k-1}. The exponential function of a planar n-complex number can be expanded in terms of the planar n-dimensional cosexponential functions f_{nk}, k=0,1,...,n-1, and expressions are given for f_{nk}. 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 \phi_k 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.