Explicit solutions of generalized Cauchy-Riemann systems using the transplant operator

Details Explicit solutions of generalized Cauchy-Riemann systems using the transplant operator Explicit solutions of generalized Cauchy-Riemann systems using the transplant operator

2009-09-30

arxiv.org/abs/0909.5665v1

Mathematics

Complex Variables

18 pages, 1 figure

Scientific paper

In [8] it was shown that the transplant operator transforms solutions of one Vekua equation into solutions of another Vekua equation, related to the first via a Schr\"odinger equation. In this paper we demonstrate a fundamental property of this operator: it transforms formal powers of the first Vekua equation into formal powers of the same order for the second Vekua equation. This property allows us to obtain positive formal powers and a generating sequence of a "complicated" Vekua equation from positive formal powers and a generating sequence of "simpler" Vekua equation. Similar results is obtained regarding construction of Cauchy kernels. Elliptic and hyperbolic pseudoanalytic function theory are considered and examples are given to illustrate the procedure.

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