Decomposition of a regular quaternion function

I. S. Polyanskii   ; V. M. Radygin   ; S. Yu. Misyurin

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 28 (2018) no. 1, pp. 36-47

Résumé

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.

Détail

Keywords: regular quaternion function, Taylor series, quaternion meromorphic function.
Mots-clés : Laurent series, residue

Bibliographie

[1] Sudbery A., “Quaternionic analyses”, Mathematical Proceedings of the Cambridge Philosophical Society, 85:2 (1979), 199–225 | DOI | MR

[2] Bitsadze A. V., Fundamentals of the theory of functions of a complex variable, Nauka, M., 1969, 240 pp.

[3] Parfenov M., “On properties of holomorphic functions in quaternionic analysis”, American Journal of Mathematical Analysis, 5:1 (2017), 17–24 | DOI

[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, 27:1 (2017), 60–68 (in Russian) | DOI

[5] Perotti A., “Regular quaternionic functions and conformal mappings”, CUBO A Mathematical Journal, 11:1 (2009), 123–143 | MR

[6] Hamilton W. R., Selected works. Optics. Dynamics. Quaternions, Nauka, M., 1994, 560 pp.

[7] Fueter R., “Über die analytische darstellung der regulären funktionen einer quaternionenvariablen”, Commentarii Mathematici Helvetici, 8:1 (1935), 371–378 | DOI | MR

[8] Gentili G., Mariconda C., Tarallo M., “Quaternionic regular maps and $\overline{\partial}$-type operators”, A Quaterly Jornal of Pure and Applied Mathematics, 67:3–4 (1993), 333–359 | MR

[9] Fokas A. S., Pinotsis D. A., “Quaternions, evaluation of integrals and boundary value problems”, Computational Methods and Function Theory, 7:2 (2007), 443–476 | DOI | MR

[10] van Lancker P., “Taylor and Laurent series on the sphere”, Complex Variables, Theory and Application: An International Journal, 38:4 (1999), 321–365 | DOI | MR

[11] Gentili G., Stoppato C., “Power series and analyticity over the quaternions”, Mathematische Annalen, 352:1 (2012), 113–131 | DOI | MR

[12] Gentili G., Sarfatti G., “The Mittag-Leffler theorem for regular functions of a quaternionic variable”, New York Journal of Mathematics, 23. (2017), 583–592 | MR

[13] Gentili G., Stoppato C., Struppa D. C., Regular functions of a quaternionic variable, Springer, Berlin–Heidelberg, 2013 | DOI | MR

[14] Kolmogorov A. N., Fomin S. V., Elements of the theory of functions and functional analysis, Fizmatlit, M., 2004, 572 pp.

[15] Vilenkin N. Ya., Special functions and the theory of group representations, Nauka, M., 1991, 576 pp.

[16] Gradshtein I. S., Ryzhik I. M., Table of integrals, sums, series and products, Fizmatlit, M., 1963, 1100 pp.

[17] Shtepina T. V., “A generalization of the Funk–Hecke theorem to the case of hyperbolic spaces”, Izvestiya: Mathematics, 68:5 (2004), 1051–1061 | DOI | DOI | MR

[18] Shabat B. V., Introduction to complex analysis, Nauka, M., 1976, 720 pp.