  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.
