Let

$$I(x)=\int_{0}^{\pi} f(t) e^{i x \psi(t)} d t$$

where $f(t)$ and $\psi(t)$ are smooth, and $\psi^{\prime}(t) \neq 0$ for $t>0 ;$ also $f(0) \neq 0$, $\psi(0)=a$, $\psi^{\prime}(0)=\psi^{\prime \prime}(0)=0$ and $\psi^{\prime \prime \prime}(0)=6 b>0$. Show that, as $x \rightarrow+\infty$,

$$I(x) \sim f(0) e^{i(x a+\pi / 6)}\left(\frac{1}{27 b x}\right)^{1 / 3} \Gamma(1 / 3) .$$

Consider the Bessel function

$$J_{n}(x)=\frac{1}{\pi} \int_{0}^{\pi} \cos (n t-x \sin t) d t$$

Show that, as $n \rightarrow+\infty$,

$$J_{n}(n) \sim \frac{\Gamma(1 / 3)}{\pi} \frac{1}{(48)^{1 / 6}} \frac{1}{n^{1 / 3}}$$