Let

$$I_{0}=\int_{C_{0}} e^{x \phi(z)} d z \text {, }$$

where $\phi(z)$ is a complex analytic function and $C_{0}$ is a steepest descent contour from a simple saddle point of $\phi(z)$ at $z_{0}$. Establish the following leading asymptotic approximation, for large real $x$ :

$$I_{0} \sim i \sqrt{\frac{\pi}{2 \phi^{\prime \prime}\left(z_{0}\right) x}} e^{x \phi\left(z_{0}\right)}$$

Let $n$ be a positive integer, and let

$$I=\int_{C} e^{-t^{2}-2 n \ln t} d t$$

where $C$ is a contour in the upper half $t$-plane connecting $t=-\infty$ to $t=\infty$, and $\ln t$ is real on the positive $t$-axis with a branch cut along the negative $t$-axis. Using the method of steepest descent, find the leading asymptotic approximation to $I$ for large $n$.