On Various Approaches to Asymptotics of Solutions to the Third Painlevé Equation in a Neighborhood of Infinity

Abstract

We examine asymptotic expansions of the third Painlevé transcendents for αδ ≠ = 0 and γ = 0 in the neighborhood of infinity in a sector of aperture <2π 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 the 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.