Ya.A. Kalinovsky, Y.E. Boyarinova, T.V. Sinkova, A.S. Sukalo

Èlektron. model. 2018, 38(3):23-32


Representations of the trigonometric functions of the generalized quaternion based on the method of associated system of differential equations have been constructed


hypercomplex number system, exponential function, trigonometric function, sinus, cosinus, generalized quaternion, basis, Cayley’s table.


1. Godel, C. (1949), An Example of a New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation, Rev. Mod. Phys., Vol. 21, no. 3, pp. 447-450.
2. Klipkov, S.I. (2014), “Generalized analysis of matrix representations of associative hypercomplex number systems used in the energy problems”, Reyestratsiya, zberigannya i obrobka danykh, Vol. 16, no. 2, pp. 28-41.
3. Alagos, Ya., Oral, K. At H. and Yuce, S. (2012), Split Quaternion Matrices, Miscolc Mathematical Notes, Vol. 13, no. 2, pp. 223-232.
4. Janovska, D. and Opfer, G. (2013), “Linear equations and the Kronecker product in coquaternions”, Mitt. Math. Ges Hamburg, Vol. 33, pp. 181-196.
5. Kalinovsky, Ya.A., Boyarinova, Yu.E. and Turenko, A.S. (2015), “Research relations between generalized quaternion and Grassmann-Clifford doubling procedures”, Reyestratsia, zberigannya i obrobka danykh, Vol. 17, no. 1, pp. 36-45.
6. Mamagami, A.B. and Jafari, M. (2013), “Some Notes on Matrix of Generalized Quaternion”, International Research Journal of Applied and Basic Sciences, Vol. 7, no. 14, pp. 1164-1171.
7. Kalinovsky, Ya.A., Turenko, A.S., Boyarinova, Yu.E. and Khitsko, Ya.V. (2015), “Properties of generalized quaternions and their relation with the Grassmann-Clifford doubling procedure”, Elektronnoe modelirovanie, Vol. 37, no. 2, pp. 17-26.
8. Boyarinova, Yu.E., Kalinovsky, Ya.A. and Sukalo, A.S. (2015), “Construction of digital signature algorithm using functions of generalized quaternions”, Reyestratsia, zberigannya i obrobka danykh, Vol. 17, no. 3, pp. 48-55.
9. Kalinovsky, Ya.A., Boyarinova, Yu.E. and Sukalo, A.S. (2015), “Mathematical modeling of representations of exponential and logarithmic functions in hypercomplex numerical system of generalized quaternions”, Reyestratsia, zberigannya i obrobka danykh, Vol. 17, no. 4, pp. 11-20.
10. Sinkov, M.V., Kalinovsky, Ya.O. and Boyarinova, Yu.E. (2010), Konechnomernye giperkompleksnye chislovye sistemy. Osnovy teorii. Primeneniya [Finite-dimensional hypercomplex number systems. Fundamentals of the theory. Applications], Institut problem registratsii informatsii NAN Ukrainy, Kyiv, Ukraine.
11. Kalinovsky, Ya.O. (2007), “Methods of computer modeling and calculations using hypercomplex number systems”, Thesis for Dr. Sci. (Tech.) degree, Institute for Problems of Information Recording of NAS of Ukraine, Kyiv, Ukraine.
12. Kalinovsky, Ya.A. (2003), “Study of isomorphism properties of quadriplex and bicomplec numerical systems”, Reyestratsia, zberigannya i obrobka danykh, Vol. 5, no. 1, pp. 69-73.
13. Kalinovsky, Ya.A., Roenko, N.V. and Sinkov, M.V. (1996), “Methods for constructing nonlinear functions in extensions of complex numbers”, Kibernetika i sistemnyi analiz, no. 4, pp. 178-181.
14. Hamilton, W.R. (1848), “Researches respecting quaternions: First series”, Transactions of the Royal Irish Academy, Vol. 21, Part 1, pp. 199-296.
15. Kàhler, U. (1998), “Die Anwendung der hyperkomplexen Funktionentheorie auf die Losung partieller Differentialgleichungen”, available at: www.tu-chemnitz.demathematik/ prom_habil/promint.pdf (1998).
16. Brackx, F. (1979), “The Exponential Function of a Quaternion Variable”, Applicable Analysis, Vol. 8, pp. 265-276.
17. Scheicher, K., Tichy, R.F. and Tomantschger, K.W. (1997), “Elementary Inequalities in Hypercomplex Numbers”, Anzeiger, Vol. II, no. 134, pp. 3-10.
18. Holin, H. “The quaternionic exponential and beyond”, available at:
19. Sinkov, M.V., Kalinovsky, Ya.A. and Sinkova, T.V. (2002), “Some linear and non-linear operation of generalized complex numbers”, Reyestratsia, zberigannya i obrobka danykh, Vol. 4, no. 3, pp. 55-61.

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