By means of Power Geometry, shortly presented in §1, in the generic case we compute all power expansions of solutions to the fifth Painleve' equation at points and z=0 and z = ∞. Exept known expansions being power series, we have found expansions with a more complicate set of power exponents. In particularly, we have found a family for which expansions begin from arbitrary power of the independent variable with arbitrary constant coefficient.

Mathematical problems and theory of numerical methods