Expansions of solutions to the fifth Painlevé equation near its nonsingular point

A. D. Bruno, A. V. Parusnikova


Abstract: The article is devoted to the study of the fifth Painlevé 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 $z=z_0$, $z_0 \ne 0$, $z_0 \ne \infty$ 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 $z - z_0$. One of them is new: it has an arbitrary coefficient of the $(z - z_0)^4$. One of these families is two-parameter, other are one-parameter. All the expansions converge near the point $z=z_0$.