Consider the integral representation for the modified Bessel function

$$I_{0}(x)=\frac{1}{2 \pi i} \oint_{C} t^{-1} \exp \left[\frac{i x}{2}\left(t-\frac{1}{t}\right)\right] d t$$

where $C$ is a simple closed contour containing the origin, taken anti-clockwise.

Use the method of steepest descent to determine the full asymptotic expansion of $I_{0}(x)$ for large real positive $x .$