On the radius of convergence of the simplest power series solution of Painlevé-I equation

Abstract

The solutions for the Painleve-I equation are expected to be new transcendental functions. But we do not have any concrete information about each solution. The simplest power series solution which is regular at the origin is considered. One of the most important features of this solution is the location of singularities. The location of the nearest singularity from the origin is given by the radiusR of convergence of this power series. The value ofR is calculated numerically by the formula of Cauchy-Hadamard and by that of d’Alembert. A theoretically correct method calculatingR is proposed. We derive some Briot-Bouquet equation from the Painleve-I equation and solve it numerically by the Runge-Kutta method. Thus we obtainedR=2.6155…. Accurate theoretical bounds forR are also obtained by various methods.