Let

$$I(\lambda, a)=\int_{-i \infty}^{i \infty} \frac{e^{\lambda\left(t^{3}-3 t\right)}}{t^{2}-a^{2}} d t$$

where $\lambda$ is real, $a$ is real and non-zero, and the path of integration runs up the imaginary axis. Show that, if $a^{2}>1$,

$$I(\lambda, a) \sim \frac{i e^{-2 \lambda}}{1-a^{2}} \sqrt{\frac{\pi}{3 \lambda}}$$

as $\lambda \rightarrow+\infty$ and sketch the relevant steepest descent path.

What is the corresponding result if $a^{2}<1$ ?