(a) Find the Stokes ray for the function $f(z)$ as $z \rightarrow 0$ with $0<\arg z<\pi$, where

$$f(z)=\sinh \left(z^{-1}\right)$$

(b) Describe how the leading-order asymptotic behaviour as $x \rightarrow \infty$ of

$$I(x)=\int_{a}^{b} f(t) e^{i x g(t)} d t$$

may be found by the method of stationary phase, where $f$ and $g$ are real functions and the integral is taken along the real line. You should consider the cases for which:

(i) $g^{\prime}(t)$ is non-zero in $[a, b)$ and has a simple zero at $t=b$.

(ii) $g^{\prime}(t)$ is non-zero apart from having one simple zero at $t=t_{0}$, where $a<t_{0}<b$.

(iii) $g^{\prime}(t)$ has more than one simple zero in $(a, b)$ with $g^{\prime}(a) \neq 0$ and $g^{\prime}(b) \neq 0$.

Use the method of stationary phase to find the leading-order asymptotic form as $x \rightarrow \infty$ of

$$J(x)=\int_{0}^{1} \cos \left(x\left(t^{4}-t^{2}\right)\right) d t$$

[You may assume that $\left.\int_{-\infty}^{\infty} e^{i u^{2}} d u=\sqrt{\pi} e^{i \pi / 4} .\right]$