The Bessel function $J_{\nu}(z)$ is defined, for $|\arg z|<\pi / 2$, by

$$J_{\nu}(z)=\frac{1}{2 \pi i} \int_{-\infty}^{\left(0^{+}\right)} \mathrm{e}^{\left(t-t^{-1}\right) z / 2} t^{-\nu-1} d t,$$

where the path of integration is the Hankel contour and $t^{-\nu-1}$ is the principal branch.

Use the method of steepest descent to show that, as $z \rightarrow+\infty$,

$$J_{\nu}(z) \sim(2 / \pi z)^{\frac{1}{2}} \cos (z-\pi \nu / 2-\pi / 4) .$$

You should give a rough sketch of the steepest descent paths.