Exponential forms and path integrals for complex numbers in n dimensions

Abstract

Two distinct systems of commutative complex numbers in n dimensions are described, of polar and planar types. Exponential forms of n-complex numbers are given in each case, which depend on geometric variables. Azimuthal angles, which are cyclic variables, appear in these forms at the exponent, and this leads to the concept of residue for path integrals of n-complex functions. The exponential function of an n-complex number is expanded in terms of functions called in this paper cosexponential functions, which are generalizations to n dimensions of the circular and hyperbolic sine and cosine functions. The factorization of n-complex polynomials is discussed.