Power expansions of solutions to the modified third Painlevé equation in a neighborhood of zero

Abstract

The modified third Painlevé equation

$$\ddot w = \frac{{\dot w^2 }}{w} - \frac{{\dot w}}{t} + \frac{{aw^2 + b}}{t} + cw^3 + \frac{d}{w}$$

, where ẇ = dw/dt and a, b, c, and d are complex parameters, is considered. Let a, b, c, d ≠ 0. The author studied asymptotic expansions of its solutions in a neighborhood of t = 0 having the form

$$w = \sum {c_k t^k } , k \in K$$

, where c _k are complex constants or polynomials in ln t with complex coefficients. All possible power-logarithmic expansions of solutions to the modified third Painlevé equation are obtained.