De-Moivre and Euler Formulae for Dual-Hyperbolic Numbers

Year 2019, Volume: 2 Issue: 3, 180 - 184, 30.12.2019

Mehmet Ali Güngör , Elma Kahramani

Abstract

In this study, we generalize the well-known formulae of de-Moivre and Euler of hyperbolic numbers to dual-hyperbolic numbers. Furthermore, we investigate the roots and powers of a dual-hyperbolic number by using these formulae. Consequently, we give some examples to illustrate the main results in this paper.

Keywords

Dual number , Hyperbolic number

References

• [1] J. Cockle On a new imaginary in algebra, London-Dublin-Edinburgh Philosophical Magazine 3(34) (1849), 37-47.
• [2] I. M. Yaglom, A simple non-Euclidean geometry and its physical basis, Springer-Verlag New York, 1979.
• [3] S. Yüce, Z. Ercan, On properties of the dual quaternions, European Journal of Pure and Applied Mathematics 4(2) (2011), 142-146.
• [4] G. Sobczyk The hyperbolic number plane, The College Math. J., 26(4) (1995), 268-280.
• [5] F. Catoni, R. Cannata, V. Catoni, P. Zampetti, Hyperbolic trigonometry in two-dimensional space-time geometry, Nuovo Cimento della Societa Italiana di Fisica B 118 (2003), 475-491.
• [6] S. Yüce, N. Kuruo˜glu , One-parameter plane hyperbolic motions, Adv. Appl. Clifford Alg. 18(2) (2018), 279-285.
• [7] M. Akar, S. Yüce, S. Sahin, On the Dual Hyperbolic Numbers and the Complex Hyperbolic Numbers, Journal of Computer Science Computational Mathematics, 8(1) (2018), 279-285.
• [8] S. Ersoy, M. Akyiğit, One-parameter homothetic motion in the hyperbolic plane and Euler-Savary formula, Adv. Appl. Clifford Alg. 21(2) (2011), 297-317.
• [9] D. P. Mandic, V. S. L. Goh, Hyperbolic valued nonlinear adaptive filters: noncircularity, widely linear and neural models, John Wiley-Sons., 2009.
• [10] G. Helzer, Special relativity with acceleration, Amer. Math. Monthy 107(3) (2000), 219-237.
• [11] W. K. Clifford, Preliminary sketch of bi-quaternions, Proc. London Math. Soc. 4 (1873), 381-395.
• [12] E. Study, Geometrie der dynamen, Leipzig, Germany, 1903.
• [13] E. Cho, De-Moivreâ˘AZ´s formula for quaternions, Appl. Math. Lett. 11(6) (1998), 33-35.
• [14] H. Kabadayı, Y. Yaylı, De-Moivre’s formula for dual quaternions, Kuwait J. Sci. Technol. 38(1) (2011), 15-23.
• [15] I. A. Kösal, A note on hyperbolic quaternions, Universal Journal Of Mathematics and Applications 1(3) (2018), 155-159.
• [16] V. Majernik, Multicomponent number systems, Acta Physics Polonica A 3(90) (1996), 491-498.
• [17] F. Messelmi, Dual-hyperbolic numbers and their holomorphic functions, (2015), https://hal.archives-ouvertes.fr/hal-01114178.