Asymptotic Properties of Special Function Solutions of the Painlevé III Equation for Fixed Parameters

In this paper, we compute the small and large $x$ asymptotics of the special function solutions of the Painlevé-III equation in the complex plane. We use the representation in terms of Toeplitz determinants of Bessel functions obtained by Masuda. Toeplitz determinants are rewritten as multiple contour integrals using Andrèief's identity. The small and large $x$ asymptotics are obtained using elementary asymptotic methods applied to the multiple contour integral. The asymptotics is extended to the whole complex plane using analytic continuation formulas for Bessel functions. The claimed result has not appeared in the literature before. We note that the Toeplitz determinant representation is useful for numerical computations of corresponding solutions of the Painlevé-III equation.