Explicit solutions of generalized Cauchy-Riemann systems using the transplant operator. (English) Zbl 1194.30052
Summary: In [V.V. Kravchenko, J. Math. Anal. Appl. 339, No. 2, 1103–1111 (2008; Zbl 1135.30020)] it was shown that the tool introduced there and called the transplant operator transforms solutions of one Vekua equation into solutions of another Vekua equation, related to the first via a Schrödinger equation. In this paper, we prove a fundamental property of this operator: it preserves the order of zeros and poles of generalized analytic functions and 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 a “simpler” Vekua equation. Similar results are obtained regarding the construction of Cauchy kernels. Elliptic and hyperbolic pseudoanalytic function theories are considered and examples are given to illustrate the procedure.
MSC:
| 30G20 | Generalizations of Bers and Vekua type (pseudoanalytic, \(p\)-analytic, etc.) |
Keywords:
Vekua equation; generalized analytic function; pseudoanalytic function; stationary Schrödinger equationCitations:
Zbl 1135.30020References:
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