Commutative hypercomplex numbers and functions of hypercomplex variable: a matrix study

Abstract

Systems of hypercomplex numbers, which had been studied and developed at the end of the 19 th century, are nowadays quite unknown to the scientific community. It is believed that study of their applications ended just before one of the fundamental discoveries of the 20 th century, Einstein's equivalence between space and time. Owing to this equivalence, not-defined quadratic forms have got concrete physical meaning and have been recently recognized to be in strong relationship with a system of bidimensional hypercomplex numbers. These numbers (called hyperbolic) can be considered as the most suitable mathematic language for describing the bidimensional space-time, in spite of some unfamiliar algebraic properties common to all the commutative hypercomplex systems with more than two dimensions: they are decomposable systems and there are non-zero numbers whose product is zero. With respect to the famous Hamilton quaternions, one can introduce the differential calculus for the hyperbolic numbers and for all the commutative hypercomplex systems; moreover, one can even introduce functions of hypercomplex variable.The aim of this work is to study the systems of commutative hypercomplex numbers and the functions of hypercomplex variable by describing them in terms of a familiar mathematical tool, i.e. matrix algebra.

Two-dimensional hypercomplex numbers and related trigonometries and geometries

“…In more recent years the bidimensional hypercomplex systems in general form have been considered in detail [1,2,15,13]. 2 Hypercomplex numbers [14,16] are defined by the expression:…”

“…The versor product defines also the product of hypercomplex numbers. This product definition makes the difference between vector algebra and hypercomplex systems and allows to relate the hypercomplex numbers to finite and infinite [16,6] Lie groups. In fact the vector product is not, in general, a vector, while the product of hypercomplex numbers is still an hypercomplex number; this is true for the division too, that for vectors does not exist while for hypercomplex numbers, in general, does exist.…”

“…As it is well known, such units (both elliptic and hyperbolic) can be decomposed into idempotent basis ui: u+, u−, where u+ = 1/2(1 + u), u− = 1/2(1 − u), for which it results: (ui) n = ui and u+ u− = 0 [4,9]. Such decomposition is very useful for the effective calculation of the functions of hypercomplex variable [15] It has been shown [1,2], depending on the sign of the real quantity…”

“…181] [18]. 6 z andz are the solutions of the characteristic equation related to the number systems [16]. 7 The definition of distances (metric element) is equivalent to introduce the bilinear form of the scalar product.…”

“…(21) the first expression of Q given in eq. (16), and the second expression of D i, j given in eq. (5), and comparing the coefficients of the left-and right-hand sides, we obtain a six equations system, whose solution is given by:…”