Advances in Applied Clifford Algebras

, Volume 8, Issue 1, pp 109–128 | Cite as

Hyperbolic Calculus

  • A. E. Motter
  • M. A. F. Rosa


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.


Riemann Surface Light Cone Conformal Transformation Clifford Algebra Stereographic Projection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Porteous I. R., “Topological Geometry”, Cambridge University Press, Cambridge, 1981.zbMATHGoogle Scholar
  2. [2]
    Fjelstad P., Extending Special Relativity via the Perplex Numbers,Am. J. Phys. 54 (5) (1986).Google Scholar
  3. [3]
    Laurentiev M. and B. Chabat, “Effects Hydrodynamiques et Modèles Mathématiques” Mir, Moscow, 1980.Google Scholar
  4. [4]
    Assis A. K. T., Perplex Numbers and Quaternions,Int. J. Math. Educ. Tech. 22 (4) (1991); G. Sobczyk, The Hyperbolic Number Plane, preprint; V. Majernik,Spec. Sci. Technol. 6 (1983) 189.Google Scholar
  5. [5]
    Keller J., Quaternionic, Complex, Duplex and Real Clifford Algebras,Adv. Appl. Clifford Alg. 4, No 1 (1994), 1.zbMATHGoogle Scholar
  6. [6]
    Delanghe R., F. Sommen, V. Soucek, F. Brackx and D. Constares, “Clifford Algebra and Spinor-Valued Functions”, Kluwer Academic Publishers, Dordrecht, 1992.zbMATHGoogle Scholar
  7. [7]
    Recami E. and R. Mignani,Lett Nuovo Cimento 4 (1972) 144.CrossRefGoogle Scholar
  8. [8]
    Apostol T. M., “Mathematical Analysis”, Addinson-Wesley Publishing Company, 1974.Google Scholar
  9. [9]
    Zeeman E. C., The Topology of Minkowski Space,Topology,6 (1967) 161.zbMATHCrossRefGoogle Scholar
  10. [10]
    Sachs R. K., and H. Wu, “General Relativity for Mathematicians”, Springer Verlag, 1977.Google Scholar
  11. [11]
    Synge J. L., “Relativity: The General Theory”, North Holland, 1960.Google Scholar
  12. [12]
    Antonuccio F., “Semi-Complex Analysis and Mathematical Physcis”, preprint.Google Scholar
  13. [13]
    Imaeda K., A New Formulation of Eletromagnetism,Nuovo Cimento B 32 (1976), 138.CrossRefADSMathSciNetGoogle Scholar
  14. [14]
    Ryan J., Cells for Harmonicity and Generalized Cauchy Integral Formulae,Proc. London Math. Soc. 60 (3) (1992), 295.Google Scholar
  15. [15]
    Rindler W., “Introduction to Special Relativity”, Oxford University Press, 1989.Google Scholar
  16. [16]
    O’Neill B., “Semi-Riemannian Geometry”, Academic Press, 1983.Google Scholar
  17. [17]
    Dubrovin B. A., A. T. Fomenko, S. P. Novikov, “Modern Geometry”, v. I Springer Verlag, 1984.Google Scholar
  18. [18]
    Gürsey N. and H. C. Tze,Ann. Phys. 128 (1980), 29.zbMATHCrossRefADSGoogle Scholar

Copyright information

© Birkhäuser-Verlag 1998

Authors and Affiliations

  • A. E. Motter
    • 1
  • M. A. F. Rosa
    • 1
  1. 1.Department of Applied Mathematics-IMECCState University at Carnpinas (UNICAMP)CarnpinasBrazil

Personalised recommendations