International Journal of Mathematics and Mathematical Sciences
Volume 25 (2001), Issue 7, Pages 429-450
doi:10.1155/S016117120102004X
Exponential forms and path integrals for complex
numbers in n dimensions
Institute of Physics and Nuclear Engineering, Tandem Laboratory, 76900 Magurele, P.O. Box MG-6, Bucharest, Romania
Received 25 August 2000
Copyright © 2001 Silviu Olariu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.