The Boutroux ansatz for the second Painleve equation in the complex domain

An asymptotic representation of the general solution of the second Painlevé equation is constructed in a sector of the complex $z$-plane. The principal term of the asymptotics is an elliptic function whose modulus and argument are functions of $\arg z$. Explicit expressions of these functions are given, and an approximation as $|z|\to\infty$ is proved for the initial Painlevé function outside a small neighborhood of its lattice of poles.