Reduction of a transmission boundary value problem for generalized analytic functions to an integral equation (Q2703911)

Generalized analytic functions of the form \(W(z) = \phi +i\psi/P\), satisfying the following equation NEWLINE\[NEWLINE\partial W / \partial{\overline z} + A(z)(W - {\overline W}) = 0,\tag{1}NEWLINE\]NEWLINE where \(P > 0\) is a given \({\mathcal C}^{1}\)-function, are considered in a bounded (or unbounded) domain \(D\). The domain \(D\) is supposed to be divided by a curve \(\Gamma\) into two parts \(D_1\), \(D_2\). It is posed the problem to find two generalized analytic functions in \(D_1\) and \(D_2\) respectively satisfying on \(\Gamma\) \({\mathbb R}\)-linear conjugation condition (or transmission condition) NEWLINE\[NEWLINE(1-\lambda)W^{+}_{1}(z) = W^{-}_{2}(z) + \lambda {\overline {W^{-}_{2}(z)}}, z\in \Gamma. NEWLINE\]NEWLINE Generalized Cauchy type integrals are constructed in terms of the fundamental solution of the generalized Cauchy-Riemann equation (1). The above problem is reduced to a singular integral equation containing such kind of integral.