Asymptotics of Solutions to the Third Painlevé Equation

We investigate the asymptotics of the third Painlevé transcendents with the parameters $\gamma=0$, $\alpha, \beta, \delta \in \mathbb{C}$, $\alpha \delta \ne 0$ in a sector with the vertex at infinity and with the opening angle not larger than 2π. By using the method of dominant balance for analyzing the third Painlevé equation, we obtain possible asymptotics of solutions to this equation expressed in terms of elliptic functions. Then we provide the recurrence relation for the coefficients of the formal asymptotic expansions of solutions to the third Painlevé transcendents into Puiseux series. Also, we present a family of values of the parameters of the third Painlevé equation such that the above Puiseux series – considered as series of $z^{2/3}$ - are series of exact Gevrey order one. We provide the analytic functions which are approximated of Gevrey order by these series in sectors with the vertices at infinity and with the opening angles not larger than 2π.