De-Moivre and Euler Formulae for Dual-Complex Numbers

Year 2019, Volume: 2 Issue: 3, 126 - 129, 30.09.2019

Mehmet Ali Güngör , Ömer Tetik

https://doi.org/10.32323/ujma.587816

Cited By: 4

Abstract

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

Keywords

Complex Number, dual numbers

References

• [1] D. P. Mandic, V. S. L. Goh, Complex valued nonlinear adaptive filters: noncircularity, widely linear and neural models, John Wiley Sons, 2009.
• [2] W. K. Clifford, Preliminary sketch of bi-quaternions, Proc. London Math. Soc., 4(1873), 381-395.
• [3] E.Study, Geometrie der Dynamen, Leipzig, Germany, 1903.
• [4] I. M. Yaglom, A Simple Non-Euclidean Geometry and Its Physical Basis, Springer-Verlag New York, 1979.
• [5] S. Y¨uce, Z. Ercan, On Properties of the Dual Quaternions, Eur. J. Pure Appl. Math., 4(2)(2011), 142-146.
• [6] G. Helzer, Special Relativity with acceleration, Amer. Math. Monthy, 107(3)(2000), 219-237.
• [7] E. Cho, De-Moivre’s formula for quaternions, Appl. Math. Lett. 11(6)(1998), 33-35.
• [8] H. Kabadayı, Y. Yaylı, De-Moivre’s formula for dual quaternions, Kuwait J. Sci. Tech, 38(1)(2011), 15-23.
• [9] I. A. K¨osal, A note on hyperbolic quaternions, Univers. J. Math. Appl., 1(3)(2018), 155-159.
• [10] V. Majernik, Multicomponent number systems, Acta Phys. Polon. A, 3(90)(1996), 491-498.
• [11] F. Messelmi, Dual-Complex Numbers and Theır Holomorphic Functions, https://hal.archives-ouvertes.fr/hal-01114178, (2015).