This paper considers the asymptotic form of solutions of the equation $y_{xx} = (u^2 - 2h^2 \cos 2x)y$ for fixed real values of x and h and large complex values of u. Attention is focused on that solution known as the Mathieu function of the third kind, $M_\nu ^{(3)} (x)$, and for values of u in the half plane $\operatorname{Re} (u) > 0$. The basic asymptotic formulas require the determination of an elliptic integral but, when u is large, it is shown how this integral can be approximated by elementary functions.