ASYMPTOTIC EXPANSIONS OF SOLUTIONSOF THE SIXTH PAINLEVE EQUATION

摘要

We obtain all asymptotic expansions of solutions of the sixth Painleve equation near all three singular points x = 0, x = 1, and x = ∞ for all values of four complex parameters of this equation. The expansions are obtained for solutions of five types: power, power-logarithmic, complicated, semiexotic, and exotic. They form 117 families. These expansions may contain complex powers of the independent variable x. First we use methods of two-dimensional power algebraic geometry to obtain those asymptotic expansions of all five types near the singular point x = 0 for which the order of the leading term is less than 1. These expansions are called basic expansions. They form 21 families. All other asymptotic equations near three singular points are obtained from basic ones using symmetries of the equation. The majority of these expansions are new. Also, we present examples and compare our results with previously known ones.