The Cauchy formula and Cauchy type integral for a class of generalized analytic functions (Q1293612)

An analogue of the Cauchy kernel for a class of generalized (in Vekua-Bers sense) analytic functions is constructed. Namely, complex-valued functions satisfying the following equation \[ 2\;\frac{\partial}{\partial {\bar z}} \Phi(z)-\frac{2m+1}{z-{\bar z}} \left (\Phi(z)-\overline{\Phi(z)}\right)=0 \tag{1} \] are considered. It is proved that the integral \[ \frac{1}{2\pi i}\int_{L} \Phi(z)w(z,\tau) d\tau, \] with \[ w(z,\tau) =\frac{\omega(z,\tau)}{z-\tau}, \] \[ \omega(z,\tau)= \frac{2m+1}{2}\left(\frac{\eta}{y}\right) |\eta|\int_{0}^{\pi}(\cos m\theta -\cos(m+1)\theta) \frac{d\theta}{\Delta}, \] \[ \tau= \xi+i\eta, \Delta=\left[(x-\xi)^{2}+y^{2}+\eta^{2}- 2y\eta\cos\theta\right]^{1/2}, \] possesses the properties of the Cauchy integral whenever \(L\) is a piecewise smooth boundary of a domain symmetric with respect to a real line, and \(\{\Phi(z)=\overline{\Phi{\bar z}}\}\) satisfying (1). Moreover, if \(f\) is Hölder-continuous on \(L\), \(\{f(\tau)=\overline{f(\bar \tau)}\}\) \((-\infty<m<\infty)\), \(f(\xi)=0\) \((m<0)\), then the integral \[ \frac{1}{2\pi i}\int_{L} f(\tau)w(z,\tau) d\tau \] possesses the properties of a Cauchy-type integral. In particular, the Sokhotsky-Plemelj formulae hold.