Power Geometry as a new mathematics

Here we present basic ideas and algorithms of Power Geometry and give a survey of some of its applications. In Section 1, we consider one generic ordinary differential equation and demonstrate how to find asymptotic forms and asymptotic expansions of its solutions. In Section 2, we demonstrate how to find expansions of solutions to Painlevé equations by this method, and we analyze singularities of plane oscillations of a satellite on an elliptic orbit. In Section 3, we expound the space generalizations of constructions of Section 1 and give some applications to Mechanics and Hydromechanics.