Painlevé property

Summary

An ordinary differential equation in the complex domain has the Painlevé property if its movable singularities are poles. Here a singularity of a solution is movable when its position depends on the initial data. Linear equations have it, since their solutions can be continued along every path that avoids the singular points of the coefficients. Nonlinear equations with the property are rare, and Painlevé and his school classified those of second order of a certain rational type. Six new equations appeared in this classification, the Painlevé equations, whose solutions define new special functions. See the definition.