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