International Journal of Mathematics and Mathematical Sciences
Volume 25 (2001), Issue 7, Pages 429-450

Exponential forms and path integrals for complex numbers in n dimensions

Silviu Olariu

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.


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.