Consider the contour-integral representation

$$J_{0}(x)=\operatorname{Re} \frac{1}{i \pi} \int_{C} e^{i x \cosh t} d t$$

of the Bessel function $J_{0}$ for real $x$, where $C$ is any contour from $-\infty-\frac{i \pi}{2}$ to $+\infty+\frac{i \pi}{2}$.

Writing $t=u+i v$, give in terms of the real quantities $u, v$ the equation of the steepest-descent contour from $-\infty-\frac{i \pi}{2}$ to $+\infty+\frac{i \pi}{2}$ which passes through $t=0$.

Deduce the leading term in the asymptotic expansion of $J_{0}(x)$, valid as $x \rightarrow \infty$

$$J_{0}(x) \sim \sqrt{\frac{2}{\pi x}} \cos \left(x-\frac{\pi}{4}\right)$$