On various approaches to asymptotics of solutions to the third Painlevé equation in a neighborhood of infinity

We examine asymptotic expansions of the third Painlevé transcendents for $\alpha \delta \ne 0$ and $\gamma=0$ in a neighborhood of infinity in a sector of aperture ${<}2 \pi$ by the method of dominant balance). We compare intermediate results with results obtained by methods of three-dimensional power geometry. We find possible asymptotics in terms of elliptic functions, construct a power series, which represents an asymptotic expansion of a solution to the third Painlevé equation in a certain sector, estimate the aperture of this sector, and obtain a recurrent relation for the coefficients of the series.