Planar complex numbers in even n dimensions
  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. 
Silviu Olariu (National Institute of Physics and Nuclear Engineering,   Tandem Laboratory, Magurele, Bucharest, Romania)