(a) State Watson's lemma for the case when all the functions and variables involved are real, and use it to calculate the asymptotic approximation as $x \rightarrow \infty$ for the integral $I$, where

$$I=\int_{0}^{\infty} e^{-x t} \sin \left(t^{2}\right) d t$$

(b) The Bessel function $J_{\nu}(z)$ of the first kind of order $\nu$ has integral representation

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

where $\Gamma$ is the Gamma function, $\operatorname{Re}(\nu)>1 / 2$ and $z$ is in general a complex variable. The complex version of Watson's lemma is obtained by replacing $x$ with the complex variable $z$, and is valid for $|z| \rightarrow \infty$ and $|\arg (z)| \leqslant \pi / 2-\delta<\pi / 2$, for some $\delta$ such that $0<\delta<\pi / 2$. Use this version to derive an asymptotic expansion for $J_{\nu}(z)$ as $|z| \rightarrow \infty$. For what values of $\arg (z)$ is this approximation valid?

[Hint: You may find the substitution $t=2 \tau-1$ useful.]