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Asymptotic behavior of the general real solution of a Painleve equation of the third kind. (English) Zbl 0674.34048

The author here considers the equations

(1)u '' +x -1 u ' +sinu=0

and gives a complete description of the asymptotic behaviour of the real solutions of (1) with the initial condition u(x)=rlnx+s+O(x 2 ) as x0. A special class of complex solutions of (1) is also identified which have no singularities for real x and which become solutions of the Hankel type

uCH 0 (x)αx -1/2 exp±i(x+β)

at infinity.

Reviewer: J.O.C.Ezeilo
MSC:
34E05Asymptotic expansions (ODE)