Hyperbolic calculus (Q1276107)

The goal of the paper is to present a hyperbolic calculus which bases on so-called hyperbolic numbers and is related to Lorentz transformations and dilatations in the two-dimensional Minkowski space-time. The set of hyperbolic numbers is defined by \(P=\{t+hx:t,x\in\mathbb{R}\}\), \(h^2=1\). One defines the hyperbolic conjugate of \(w=t+hx\) by \(\overline w=t-hx\) and \(\| w\|^2_M =t^2-x^2\). One denotes \(w\) time-like if \(\| w\|^2_M>0\), light-like if \(| w|^2_M=0\) and space-like if \(\| w\|^2_M<0\). This classification represents a basis to define hyperbolic Cauchy-Riemann conditions, hyperbolic derivatives, integrals, conformal transformations and so on. The authors announce a connection between hyperbolic holomorphic functions and solutions of the wave equation. But the authors did not clarify which news does hyperbolic calculus bring for the theory of the wave equations.