Asymptotic behaviour and expansions of solutions of an ordinary differential equation

Series Expansions of Painlevé Transcendents near the Point at Infinity

“…A kind of formal solutions obtained by power geometry (cf. [5]) also correspond to our solutions of these types.…”

Expansions of solutions to the fifth Painlevé equation near its nonsingular point

“…The vertices and are associ ated with truncated algebraic equations. They do not give solutions according to Remark 1.1 in [1].…”

“…The cone of the problem = {k > 0} contains only the root k = 3, which is the only critical value. The consistency condition [1] holds true. Thus, we obtain the following families of AES for Eq.…”

“…Equation (1) has two sin gular points: z = 0 and z = ∞. By applying methods of two dimensional power geometry [1,2], we find all asymptotic expansions of solutions (AES) to Eq. (1) near its nonsingular point z = z 0 , z 0 ≠ 0, z 0 ≠ ∞ for all values of the parameters α, β, γ, and δ.…”

“…Consider the fifth Painlevé equation (1) where α, β, γ, and δ are complex parameters; z and w are the respective independent and dependent com plex variables; and w' = . Equation (1) has two sin gular points: z = 0 and z = ∞.…”