Power and Exponential Expansions of Solutions to the Third Painleve Equation.
By means of Power Geometry, shortly presented in §1, in the generic case we compute all power expansions of solutions to the third Painleve equation at points z = 0 (§2) and z = z0 ≠ 0 (§3). Analogously we compute all power expansions of solutions to the modified third Painleve equation at points t = 0, t = ∞ (§4), t = t0 ≠ 0 (§5), where t = exp(z). In the point t=0 we have found a new type singularity of the modified third Painleve equation.
Mathematical problems and theory of numerical methods