Hargrave and Sleeman (1977) give asymptotic approximations for Lamé polynomials and their eigenvalues, including error bounds. The approximations for Lamé polynomials hold uniformly on the rectangle $0\le \mathrm{\Re}z\le K$, $0\le \mathrm{\Im}z\le {K}^{\prime}$, when $nk$ and $n{k}^{\prime}$ assume large real values. The approximating functions are exponential, trigonometric, and parabolic cylinder functions.