Some equations of the Fuchs class in hydro- and aeromechanics (Q1317759)

The authors consider the linear differential equation of the Fuchs class with four singular points \[ y''+ \left({1- \alpha\over \xi}+ {1- \beta\over \xi-1}+ {1-\gamma\over \xi- e}\right) y'+ {(\varepsilon\varepsilon' \xi+ \lambda)y\over \xi(\xi-1) (\xi- e)}= 0. \] This equation corresponds to the problem of the conformal mapping of the upper half-plane \(\xi\) onto some circular quadrangle of the plane \(z\) with angles \(\pi\alpha\), \(\pi\beta\), \(\pi\gamma\) and \(\pi\delta\) for which we have the Fuchs relation \(\alpha+ \beta+\gamma+ \varepsilon+\varepsilon'= 2\), \(\varepsilon- \varepsilon'= \delta\), where the affix of the one of the vertices of the quadrangle \(e\) and the so- called accessory parameter \(\lambda\) are unknown constants. In the paper it is shown that, in the whole series of the above-mentioned classes in which the polygon has angles that are multiples of \(2\pi/2\) and branch cuts, the problem of determining the unknown parameters can be completely solved.