On singular integral equations with translations (Q2741570)

The singular integral equation with constant coefficients and translations NEWLINE\[NEWLINEa\varphi(t)+\frac{b}{\pi i}\int_{-\infty}^{\infty} \frac{\varphi(\tau)}{\tau-t} d\tau + \sum_{k=1}^{n} \left(\frac{c_k}{2\pi i}\int_{-\infty}^{\infty} \frac{\varphi(\tau)}{\tau-t-\alpha_k} d\tau - \frac{d_k}{2\pi i}\int_{-\infty}^{\infty} \frac{\varphi(\tau)}{\tau-t-\beta_k} d\tau\right) =f(t),NEWLINE\]NEWLINE \(-\infty<t<\infty\), is considered. By using the method of analytic continuation this equation is reduced to a boundary value problem with special coefficients. The solution of the latter problem is presented in a closed form. NEWLINENEWLINENEWLINEReviewer's remark: The author may consult the known monographs by \textit{F. D. Gakhov} [Boundary value problems (1977; Zbl 0449.30030)], by \textit{F. D. Gakhov} and \textit{Yu. I. Cherskii} [Equations of convolution type (1978; Zbl 0458.45002)], by \textit{G. S. Litvinchuk} [Solvability theory of boundary value problems and singular integral equations with shift (2000; Zbl 0980.45001)], as well as the survey paper by \textit{I. Gokhberg} and \textit{M. G. Krein} [Am. Math. Soc., Transl., II. Ser. 14, 217-287 (1960; Zbl 0098.07501)].