Expansions of solutions to the fifth Painlev´e
equation near its nonsingular point
The article is devoted to the study of the fifth Painlev´e equation which has 4 complex parameters. By methods of Power Geometry we look for asymptotic
expansions of solutions to the equation near its nonsingular point z=z0, z0≠0, z0≠∞ for all values of parameters of the equation. We have proved that there exist exactly 10 families of expansions. These families are power series in the local variable z − z0. One of them is new: it has an arbitrary coefficient of
the (z − z0)4. One of these families is two-parameter, other are one-parameter.
All the expansions converge near the point z=z0.
Mathematical problems and theory of numerical methods