This article is cited in 5 papers

Power and exponential expansions of solutions to the third Painlevé equation

A. D. Bruno, A. V. Gridnev


Abstract: By means of Power Geometry, shortly presented in § 1, in the generic case we compute all power expansions of solutions to the third Painlevé equation at points $z=0$ (§ 2) and $z=z_0\ne0$ (§ 3). Analogously we compute all power expansions of solutions to the modified third Painlevé equation at points $t=0$, $t=\infty$ (§ 4), $t=t_0\ne0$ (§ 5), where $t=\exp(z)$. In the point $t=0$ we have found a new type singularity of the modified third Painlevé equation.