(a) Find the curves of steepest descent emanating from $t=0$ for the integral

$$J_{x}(x)=\frac{1}{2 \pi i} \int_{C} e^{x(\sinh t-t)} d t$$

for $x>0$ and determine the angles at which they meet at $t=0$, and their asymptotes at infinity.

(b) An integral representation for the Bessel function $K_{\nu}(x)$ for real $x>0$ is

$$K_{\nu}(x)=\frac{1}{2} \int_{-\infty}^{+\infty} e^{\nu h(t)} d t \quad, \quad h(t)=t-\left(\frac{x}{\nu}\right) \cosh t$$

Show that, as $\nu \rightarrow+\infty$, with $x$ fixed,

$$K_{\nu}(x) \sim\left(\frac{\pi}{2 \nu}\right)^{\frac{1}{2}}\left(\frac{2 \nu}{e x}\right)^{\nu}$$