  A system of commutative complex numbers in 5 dimensions of the form
u=x_0+h_1x_1+h_2x_2+h_3x_3+h_4x_4 is described in this paper, the variables
x_0, x_1, x_2, x_3, x_4 being real numbers. The operations of addition and
multiplication of the 5-complex numbers introduced in this work have a
geometric interpretation based on the the modulus d, the amplitude \rho, the
polar angle \theta_+, the planar angle \psi_1, and the azimuthal angles
\phi_1,\phi_2. The exponential function of a 5-complex number can be expanded
in terms of polar 5-dimensional cosexponential functions g_{5k}(y),
k=0,1,2,3,4, and the expressions of these functions are obtained from the
properties of the exponential function of a 5-complex variable. Exponential and
trigonometric forms are obtained for the 5-complex numbers, which depend on the
modulus, the amplitude and the angular variables. The 5-complex functions
defined by series of powers are analytic, and the partial derivatives of the
components of the 5-complex functions are closely related. The integrals of
5-complex functions are independent of path in regions where the functions are
regular. The fact that the exponential form of the 5-complex numbers depends on
the cyclic variables \phi_1, \phi_2 leads to the concept of pole and residue
for integrals on closed paths. The polynomials of 5-complex variables can be
written as products of linear or quadratic factors.

Complex Variables