Hyperbolic numbers and analytic functions

Abstract

In this part of the book we use the results and follow the exposition from [80] where a hyperbolic analogue of pseudoanalytic function theory was developed which proves to be extremely useful for studying hyperbolic partial differential equations. We show that solutions of the Klein-Gordon equation with an arbitrary potential are closely related to certain hyperbolic pseudoanalytic functions, the result of a factorization of the Klein-Gordon operator with the aid of two Vekua-type operators. As one of the corollaries we obtain a method for explicit construction of infinite systems of solutions of the considered Klein-Gordon equation. Our approach is based on the application of the algebra of hyperbolic numbers [110, 114] instead of that of complex numbers and generalizes some earlier works dedicated to hyperbolic analytic function theory [84, 95, 52]. It should be mentioned that the elliptic and hyperbolic pseudoanalytic function theories naturally are quite different. Nevertheless as we show following [80] there are many important common features. © 2009 Birkhäuser Verlag AG.