On a transplant operator and explicit construction of Cauchy-type integral representations for p-analytic functions

Abstract

We present a new technique for explicit construction of Cauchy kernels and Cauchy integral representations for a class of generalized analytic functions and p-analytic functions.

On the two-dimensional stationary Schrödinger equation with a singular potential

“…As was shown in [11,12] the constructed pseudoanalytic formal powers can be transformed into formal powers corresponding to other related Vekua equations preserving the order of the formal power. This substantially enlarges the class of explicitly solvable Vekua equations.…”

Cauchy integral formula for generalized analytic functions in hydrodynamics

“…The theories of p-analytic functions and generalized analytic functions defined by (1.9) furnish general forms for the Cauchy integral formula, which often need to be specialized and refined for particular classes of generalized analytic functions (Chemeris 1995;Kravchenko 2008;Zabarankin 2008a). For example, the theory of p-analytic functions facilitated obtaining the Cauchy integral formula for zero-order H -analytic functions (Zabarankin 2010).…”

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

“…In this section we define and study the main tool of this paper called the transplant operator. It was introduced in [8] and used for constructing Cauchy kernels and Cauchy integral representations for an important subclass of p-analytic functions,the x k -analytic functions. Let us describe the main idea behind this concept.…”

“…In the present paper we substantially extend the class of Vekua equations and of systems desribing p-analytic functions for which a generating sequence and a system of formal powers can be constructed explicitly. For this we use a concept introduced in [8] and called there the transplant operator. In fact, it is an operator transforming solutions of one Vekua equation into solutions of another one related to the first via a Schrödinger equation.…”

“…

In [8] 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.

…”