MATHEMATICS

Decomposition of a regular quaternion function

I. S. Polyanskii^a, V. M. Radygin^a, S. Yu. Misyurin<sup>b

a</sup> The Academy of Federal Security Guard Service of the Russian Federation, ul. Priborostroitel'naya, 35, Orel, 302034, Russia
^b National Engineering Physics Institute "MEPhI", Kashirskoe sh., 31, Moscow, 115409, Russia

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

Received: 12.10.2017

DOI: 10.20537/vm180104

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