S.I. Klipkov

Èlektron. model. 2019, 41(6):15-36


The properties of the polygenic functions of the complex and quaternionic variables used in the problems of the electric power industry are studied from the point of view of differential calcu-lus. Based on the introduced concept of internal pseudo-derivative, expressions are obtained for the right and left quotients of the differential of an arbitrary quaternion function to the dif-ferential of its argument. The conditions for the differentiability of functions of a quaternionic variable are formulated. An example of a quaternion analysis of the modes of a simple two-node circuit of an electrical system is given.


parametric hypercomplex numerical system, pseudo-derivative, complex num-bers, quaternions, quadriplex numbers.


1. Sekene, Y. and Yokojama, A. (1981), “Multisolutions for load flow problem of power System and their physical stability”, Power Systems Computer Conference, Proceeding of the 7th Power Syst. Comput. Conf, Lausanne, pp. 819-826.
2. Venikov, V.A. (1985), Perekhodnyye elektromekhanicheskiye protsessy v elektricheskikh sistemakh [Transient electromechanical processes in electrical systems], Vysshaya shkola, Moscow, USSR.
3. Kasner, E. (1936), “A complete characterization of the derivative of a polygenic func-tion”, Proceedings of the National Academy of Sciences, Vol. 22, pp. 172-177.
4. Klipkov, S.I. (2012), “The use of hypercomplex numerical systems for mathematical modeling of limiting modes of electrical systems”, Reyestratsiya, zberihannya i obrobka danykh, Vol. 14, no. 4, pp. 11-23.
5. Klipkov, S.I. (2015), “Features of harmonic analysis of limiting modes of electrical sys-tems”, Elektronnoye modelirovaniye, Vol. 37, no. 1, pp. 113-127.
6. Klipkov, S.I. (2013), “Hypercomplex parametric numerical systems in mathematical mod-eling”, Reyestratsiya, zberihannya i obrobka danykh, Vol. 15, no. 1, pp. 3-13.
7. Klipkov, S.I. (2011), “On a new approach to constructing hypercomplex numerical sys-tems of rank two over the field of complex numbers”, Ukrainskiy matematicheskiy zhurnal, Vol. 63, no. 1, pp. 130-139.
8. Sin’kov, M.V., Boyarinova, Yu.Ye. and Kalinovskiy, Ya.A. (2010), Konechnomernyye giperkompleksnyye chislovyye sistemy. Osnovy teorii. Primeneniya [Finite-dimensional hypercomplex numerical systems. Fundamentals of the theory. Applications], Infodruk, Kyiv, Ukraine.
9. Jefremov, A.P. (2004), “Quaternions: Algebra, Geometry and Physical Theories”, Giper-kompleksnyje chisla v geometriji i fizike, no. 1, pp. 111-127.
10. Sadberi, A. (2004), “Quaternion Analysis”, Giperkompleksnyje chisla v geometriji i fizike, no. 2, pp. 130-157.
11. Kassandrov, V.V. (2006), “Quaternion Analysis and Algerodynamics”, Giper-kompleksnyje chisla v geometriji i fizike, no. 2, pp. 58-84.
12. Petrov, A.M. (2006), Kvaternionnoje predstavlenije vikhrevykh dvizheniy [Quaternion rep-resentation of vortex motions], Kompanija Sputnik+, Moscow, Russia.
13. Riman, B. (1948), Sochineniya [Works], Gostekhizdat, Moscow, USSR.
14. Karpenko, I.I., Tyshkevich, D.L. and Sukhtajev, A.I. (2004), ”On an approach to the dif-ferentiation of functions of a quaternion variable”, Uchenyye zapiski TNU. Matematika, Vol. 17 (56), no. 1, pp. 30-37.
15. Oshorov, B.B. (2007), “On a four-dimensional analogue of the Cauchy-Riemann system of equations”, Neklassicheskiye uravneniya matematicheskoy fiziki. Tr. mezhdunar. konf. «Differentsial'nyye uravneniya, teoriya funktsiy i prilozheniya», pp. 212-220.
16. Shcherbina Ju.V., Zaderey A.V., Klipkov S.I. (1984), “About one method of studying the existence of steady-state modes of electrical systems”, Elektronnoye modelirovaniye, Vol. 6, no. 5, pp. 61-64.

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