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Asymptotic behavior of the general real solution of a Painleve equation of the third kind. (English. Russian original) Zbl 0674.34048
The author here considers the equations $$ (1)\quad u''+x\sp{-1}u'+\sin u=0 $$ and gives a complete description of the asymptotic behaviour of the real solutions of (1) with the initial condition $u(x)=r \ln x+s+O(x\sp 2)$ as $x\to 0$. 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 $$ u\sim CH\sb 0(x)\to \alpha x\sp{- 1/2}\exp \pm i(x+\beta) $$ at infinity.
Reviewer: J.O.C.Ezeilo
34E05Asymptotic expansions (ODE)