Closed-form integration of some Fuchsian-class differential equations found in hydro- and aeromechanics (Q1048547)

The importance of the construction of conformal mappings of the upper half plane onto a circular polygons for problems in hydro- and aeromechanics leads to the necessity of studying solutions of Fuchsian linear differential equations. If the number of singular points \(N\) of such equation is larger than three, then the research represents big difficulties. The author shows that among the equations with number of singular points \(N>3\) there exists equations which have solutions being linear combinations of solutions of the equations with smaller number of singular points. Basing on this idea, the author proves, for example, that the equation \[ v'' + \left[ {\frac{1} {{2\varsigma }} + \frac{{1 - v}} {{\varsigma - 1}} - \frac{1} {{\varsigma - {a_1}}}} \right]v' + \frac{{\tfrac{1} {4}\nu (1 + \nu )\varsigma + {\lambda _0}}} {{\varsigma (\varsigma - 1)(\varsigma - {a_1})}}v = 0 \] possesses such property, and it can be solved by means of elementary functions, if \[ \lambda _0^2 + 2{\lambda _0}[({\nu ^2} - \nu - 1){a_1} + 1] + \nu (1 + \nu ){a_1}[(1 - \nu )(2 - \nu ){a_1} - 2] = 0. \]