Complex Numbers in n Dimensions

TL;DR

This monograph explores hypercomplex numbers across various dimensions, detailing their algebraic, geometric, and analytic properties, including exponential forms, residues, and polynomial factorization in n-dimensional spaces.

Contribution

It provides a comprehensive analysis of hypercomplex systems in multiple dimensions, introducing new exponential forms, geometric interpretations, and residue concepts for n-complex functions.

Findings

Exponential forms depend on geometric variables and cyclic azimuthal angles.

Introduction of cosexponential functions as generalizations of sine and cosine.

Factorization of n-complex polynomials is discussed in detail.

Abstract

This monograph presents a detailed analysis of hypercomplex numbers in 2, 3 and 4 dimensions, then presents the properties of hypercomplex numbers in 5 and 6 dimensions. It continues with a detailed analysis of hypercomplex numbers in n dimensions, and two distinct systems of commutative complex numbers 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. The essence of this monograph is the interplay between the algebraic, the geometric and the analytic facets of the relations.