Polar complex numbers in n dimensions
  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 \rho, by 2 polar angles \theta_+,\theta_-, by n/2-2 planar angles \psi_{k-1}, and by n/2-1 azimuthal angles \phi_k. In an odd number of dimensions, the polar n-complex number can be represented by d, \rho, by 1 polar angle \theta_+, by (n-3)/2 planar angles \psi_{k-1}, and by (n-1)/2 azimuthal angles \phi_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 \phi_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. 
Silviu Olariu (National Institute of Physics and Nuclear Engineering,   Tandem Laboratory, Magurele, Bucharest, Romania)