  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.
