Pseudoanalytic functions are a generalization of ordinary analytic functions developed by I. G. Petrovskiĭ [Usp. Mat. Nauk., N. Ser. 1, No. 3–4, 44–70 (1946; Zbl 0061.20402)], M. A. Lukomskaya [Mat. Sb., N. Ser. 29(71), 551–558 (1951; Zbl 0044.09403)], S. Agmon and L. Bers [Proc. Am. Math. Soc. 3, 757–764 (1952; Zbl 0047.32101)],…. Other types of pseudoanalytic functions were introduced by G. N. Polozhiĭ [Theory and applications of \({\mathfrak p}\)-analytic and \(({\mathfrak p},{\mathfrak q})\)-analytic functions (Russian) (Kiev) (1973; Zbl 0257.30040)].
A function \(f=u+iy\) of the complex variable \(z=x+iy\) is \({\mathfrak p}\)-analytic (\(({\mathfrak p},{\mathfrak q})\)-analytic) iff \[ {\mathfrak p}u_ x=v_ y,\quad {\mathfrak p}u_ y=-v_ x, \qquad ({\mathfrak p}u_ x-{\mathfrak q}u_ y=v_ y,\;{\mathfrak q}u_ x+{\mathfrak p}u_ y=-v_ x), \] where \({\mathfrak p}\) and \({\mathfrak q}\) are positive, weakly differentiable functions.
The paper under review is closely connected to problems posed by G. N. Polozhiĭ and devoted to the generalization of the Cauchy integral for \(({\mathfrak p},{\mathfrak q})\)-analytic functions: \[ \Phi(z)= \frac{1}{2\pi i} \int_ \Gamma \varphi_1(\zeta)\,d\tilde\Omega(\zeta,z)+i\varphi_2(\zeta)\,d\Omega(\zeta,z), \] where \(\tilde\Omega(\zeta,z)\), \(\Omega(\zeta,z)\) are kernels in the domain \(E\subset G\), \(G\) is a subset of the complex plane \(\mathbb{C}\), \(\varphi_ 1+i\varphi_ 2\in C(\Gamma)\), \(\Gamma\) (\(\subset E\)) is an arbitrary Jordan curve.