Institute of Mathematics RAS,

112 Chernyshevskii Str.,

Ufa 450077,

RUSSIA

E-mail: novokshenov@yahoo.com, novik@anrb.ru

**Asymptotics of the third Painleve transcendent in the
complex plane**

**Abstract:**

An uniform asymptotics in the complex plane for the third Painleve
transcendent is constructed. Exact formulas for asymptotic parameters (the
module and the phase shift) of the elliptic function in the leading term
of the asymptotics in terms of the initial data at zero are given. Using
the isomonodromic deformation method, an analog of the Bolibrukh-Its-Kapaev
theorem for the approximation of the solution of the corresponding Riemann
problem on the complex sphere is proved.