Cauchy-Like Integral Formula for Functions of a Hyperbolic Variable

Abstract

It has been recently shown that, working in a plane with the geometry associated with hyperbolic numbers, a complete "Euclidean" formalization of geometry and trigonometry of Minkowski space-time has been obtained.In this paper we show a first result that indicates that also for the studies of functions of a hyperbolic variable, the more appropriate way is to work in a plane with pseudo-Euclidean metric.In particular it is shown that a theorem equivalent to the Cauchy integral formula for the functions of a complex variable holds for the functions of a hyperbolic variable studied in the pseudo-Euclidean plane.

The Initial Value Problem for Wave Equation and a Poisson-like Integral in Hyperbolic Plane

“…We define Q and Q * as symmetric with respect to hyperbola γ, by extending to hyperbolas in the hyperbolic plane, the Euclidean symmetry with respect to a circle. Actually these points, as it is shown in [4], have equivalent properties:…”

“…In hyperbolic plane the locus of points at the same distance from a given point is the equilateral hyperbola, then in [4] the circle with radius r → 0 has been replaced, by an equilateral hyperbola with semi-diameter ρ → 0. This means that the hyperbola becomes the parallel to axes bisectors from the point Q.…”

“…By means of this approach, in [2] the bases for studying the functions of a hyperbolic variable have been set and, for these functions, a Cauchy-like integral formula has been stated [4].…”

Some product type hyperbolic Young functions

“…Hyperbolic numbers were used in studying various other areas of mathematics and physics such as in function theory, Fourier transformation, relativistic quantum physics and many more. Many papers has appeared studying hyperbolic numbers from various points of view (see, [5], [22], [6], [11] , [13]) and references therein. During the past several years focus has been on developing hyperbolic numbers as an affordable replacement for the real number system.…”

Orlicz Spaces with Bicomplex Scalars

“…During the past several years research in this area has been to develop hyperbolic numbers as an affordable replacement for the real number system. Many papers has appeared studying hyperbolic numbers from various points of view (see, [5], [30], [6], [18] , [20]) and references therein. However in recent paper [14] studied this system of numbers as the only(natural) generalization of real numbers, into Archimedean f-algebra of dimension two.…”