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## Archive of Issues

Russia Moscow
Year
2018
Volume
28
Issue
1
Pages
36-47
 Section Mathematics Title Decomposition of a regular quaternion function Author(-s) Polyanskii I.S.a, Radygin V.M.a, Misyurin S.Yu.b Affiliations The Academy of Federal Security Guard Service of the Russian Federationa, National Research Nuclear University MEPhIb Abstract This article deals with the tasks associated with the decomposition of a regular quaternion function into generalized Taylor and Laurent series. The generalized Taylor series for a regular quaternion function were obtained by the decomposition of the Cauchy kernel in a 4-dimensional hyperball in the algebra of quaternions and the hyperspherical coordinate system. The generalized Laurent series for a regular quaternion function were obtained by the decomposition of the Cauchy kernel in the exterior of a 4-dimensional hyperball in the algebra of quaternions and the hyperspherical coordinate system. On the basis of the obtained solutions by considering the decomposition of a regular quaternion function in an infinitely small ball that is restricted by the 3-sphere, we set the rule to determine the deduction of a regular quaternion function in the algebra of quaternions and the hyperspherical coordinate system regarding the isolated singular point. In addition, the decomposition of a meromorphic quaternion function into the power series was found. Keywords regular quaternion function, Taylor series, Laurent series, residue, quaternion meromorphic function UDC 517.554 MSC 30B10 DOI 10.20537/vm180104 Received 12 October 2017 Language Russian Citation Polyanskii I.S., Radygin V.M., Misyurin S.Yu. Decomposition of a regular quaternion function, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 1, pp. 36-47. References Sudbery A. Quaternionic analyses, Mathematical Proceedings of the Cambridge Philosophical Society, 1979, vol. 85, issue 2, pp. 199-225. DOI: 10.1017/S0305004100055638 Bitsadze A.V. Osnovy teorii funktsii kompleksnogo peremennogo (Fundamentals of the theory of functions of a complex variable), Moscow: Nauka, 1969, 240 p. Parfenov M. On properties of holomorphic functions in quaternionic analysis, American Journal of Mathematical Analysis, 2017, vol. 5, no. 1, pp. 17-24. DOI: 10.12691/ajma-5-1-4 Radygin V.M., Polyanskii I.S. Methods of conformal mappings of polyhedra in $\mathbb{R}$$3$ , Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2017, vol. 27, issue 1, pp. 60-68 (in Russian). DOI: 10.20537/vm170106 Perotti A. Regular quaternionic functions and conformal mappings, CUBO A Mathematical Journal, 2009, vol. 11, no. 1, pp. 123-143. Hamilton W.R. Izbrannye trudy. Optika. Dinamika. Kvaterniony (Selected works. Optics. Dynamics. Quaternions), Moscow: Nauka, 1994, 560 p. Fueter R. Über die analytische darstellung der regulären funktionen einer quaternionenvariablen, Commentarii Mathematici Helvetici, 1935, vol. 8, issue 1, pp. 371-378. DOI: 10.1007/BF01199562 Gentili G., Mariconda C., Tarallo M. Quaternionic regular maps and $\overline{\partial}$-type operators, A Quaterly Jornal of Pure and Applied Mathematics, 1993, vol. 67, no. 3-4, pp. 333-359. Fokas A.S., Pinotsis D.A. Quaternions, evaluation of integrals and boundary value problems, Computational Methods and Function Theory, 2007, vol. 7, no. 2, pp. 443-476. DOI: 10.1007/BF03321657 van Lancker P. Taylor and Laurent series on the sphere, Complex Variables, Theory and Application: an International Journal, 1999, vol. 38, no. 4, pp. 321-365. DOI: 10.1080/17476939908815173 Gentili G., Stoppato C. Power series and analyticity over the quaternions, Mathematische Annalen, 2012, vol. 352, issue 1, pp. 113-131. DOI: 10.1007/S00208-010-0631-2 Gentili G., Sarfatti G. The Mittag-Leffler theorem for regular functions of a quaternionic variable, New York Journal of Mathematics, 2017, vol. 23, pp. 583-592. Gentili G., Stoppato C., Struppa D.C. Regular functions of a quaternionic variable, Berlin-Heidelberg: Springer, 2013. DOI: 10.1007/978-3-642-33871-7 Kolmogorov A.N., Fomin S.V. Elementy teorii funktsii i funktsional'nogo analiza (Elements of the theory of functions and functional analysis), Moscow: Fizmatlit, 2004, 572 p. Vilenkin N.Ya. Spetsial'nye funktsii i teoriya predstavleniya grupp (Special functions and the theory of group representations), Moscow: Nauka, 1991, 576 p. Gradshtein I.S., Ryzhik I.M. Tablitsy integralov, summ, ryadov i proizvedenii (Table of integrals, sums, series and products), Moscow: Fizmatlit, 1963, 1100 p. Shtepina T.V. A generalization of the Funk-Hecke theorem to the case of hyperbolic spaces, Izvestiya: Mathematics, 2004, vol. 68, no. 5, pp. 1051-1061. DOI: 10.1070/IM2004v068n05ABEH000508 Shabat B.V. Vvedenie v kompleksnyi analiz (Introduction to complex analysis), Moscow: Nauka, 1976, 720 p. Full text