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Hyperbolic Calculus
Authors:A E Motter  M A F Rosa
Institution:(1) Department of Applied Mathematics-IMECC, State University at Carnpinas (UNICAMP), CP 6065, 13081-970 Carnpinas, SP, Brazil
Abstract:The complex numbers are naturally related to rotations and dilatations in the plane. In this paper we present the function theory associate to the (universal) Clifford algebra forIR 1,0 1], the so called hyperbolic numbers 2,3,4], which can be related to Lorentz transformations and dilatations in the two dimensional Minkowski space-time. After some brief algebraic interpretations (part 1), we present a “Hyperbolic Calculus” analogous to the “Calculus of one Complex Variable”. The hyperbolic Cauchy-Riemann conditions, hyperbolic derivatives and hyperbolic integrals are introduced on parts 2 and 3. Then special emphasis is given in parts 4 and 5 to conformal hyperbolic transformations which preserve the wave equation, and hyperbolic Riemann surfaces which are naturally associated to classical string motions.
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