Application of the generalized degree method for constructing solutions of the quaternion variant of the Cauchy -- Riemann system

E. A. Loshkareva ; Yu. A. Gladyshev ; E. N. Malyshev

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 23 (2023) no. 1, pp. 11-23

Résumé

This article indicates one of the ways to solve the generalized Cauchy – Riemann system for quaternionic functions in an eight-dimensional space. In previous works, some classes of solutions of this system were studied and it was stated that it is possible to use the method of generalized degrees to construct solutions of this system of differential equations. It is shown that the solution of the problem can be reduced to finding two arbitrary quaternionic harmonic functions in an eight-dimensional space. All $8$ components of these functions $\varphi ,\psi$ must be harmonic functions, that is, be twice continuously differentiable over all $8$ real variables $x_i$, $y_i$, where $i = \overline {1,4} $ solutions of the Laplace equation. In this article, the parametric method of generalized degrees is considered, which is applicable to individual equations of the second and higher orders.