On existence and properties of roots of Third Painlevé transcendents

Abstract Separate consideration of properties of roots of Third Painlevé transcendents (P III - functions) is necessary in view of irregularity of the differential equation defining them on the subset of the phase space where its solution would vanish. Applying the Hamiltonian formalism, the mentioned second order differential equation can be replaced by two independent systems of two nonlinear first order equations whose structures allow to name them coupled Riccati equations. The existence of analytic P III - functions vanishing at a given non-zero point then directly follows. The set Z 2 × C (or Z 2 × R ) is to be applied for their indexing. It also proves to be natural to use as an unknown the third order derivative rather than the original unknown itself. After the transformation of the corresponding differential equations to equivalent integral equations the efficient algorithm of constructing of approximate solutions applicable in vicinity of their non-zero root is obtained. An example of its application is given, its numerical validation presenting results in a graphical form is carried out. The associated approximation applicable in vicinity of a pole of P III - function is given as well. The bounds from below for the distances between a pair of roots of a P III - function as well as between a root and a pole are found.