On a method for constructing generalized Bers powers in a complex space

Yu. A. Gladyshev

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the 20 International Saratov Winter School "Contemporary Problems of Function Theory and Their Applications", Saratov, January 28 — February 1, 2020. Part 2, Tome 200 (2021), pp. 36-44

Résumé

In this paper, we extend the method of generalized powers proposed by L. Bers for one variable to systems of differential equations in spaces of any number of variables, including complex variables. After a brief historical background on the issue, we formulate a general definition of generalized powers. We construct generalized powers for two complex variables $z$ and $\overline{z}$; this construction is a prototype for the multidimensional case. We construct solutions of a three-dimensional generalization of the Cauchy–Riemann system and the corresponding Laplace equations. We introduce binary generalized powers that are analogs of complex powers of the form $z^n\overline{z}^m c$. These structures provides a possibility of generalizing the method of generalized powers to the four-dimensional case, which is important in physics. We show that the techniques of generalized powers can be used for constructing solutions of the Maxwell equations in the classical field theory and the Dirac equations in the quantum theory of electrons.

Détail

Keywords: complex space, generalized power, Cauchy-Riemann system.