Using Power Geometry [1,2],in the generic case we find all expansion of solutions to the sixth Painleve' equation [3,4]near a nonsingular point of the independent variable, i.e. different from zero, one and infinity. All expansions contain integral power exponents of the local variable and have constant complex coefficients and converge. There are 5 families of such expansions. Expansions of solutions near the singular points are described in [2,5].

Mathematical problems and theory of numerical methods