All expansions of solutions to the sixth Painlevé equation near its nonsingular point

A. D. Bruno, I. V. Goryuchkina


Abstract: Here we consider the sixth Painlevé equation for all values of four its complex parameters $a,b,c,d$ near its nonsingular point $x=x_0\ne 0,1,\infty$ and we look for all asymptotic expansions of its solutions of four types: power, power-logarithmic, complicated, exotic and also exponential asymptotic forms. Altogether they form 17 families and all of them are power. Expansions of other three types and exponent asymptotic forms are absent, as it must be for a Painlevé equation. Eight of these 17 families are new. The other 9 families were known.