Expansions of solutions to the sixth Painlevé equation near singular points $x=0$ и $x=\infty$

A. D. Bruno, I. V. Goryuchkina


Abstract: We consider the sixth Painlevé equation in the case $a,b\ne0$. By the methods of Power Geometry, near the singular points $x=0$ and $x=\infty$, we have found all power, power-logarithmic and complicated expansions of its solutions. Near $x=0$ we have obtained 15 families of expansions, sixth of them are complicated. Using a symmetry of the equation, near $x=\infty$ we have obtained again 15 families of expansions, including 6 complicated.