What is meant by the asymptotic relation

$$f(z) \sim g(z) \quad \text { as } \quad z \rightarrow z_{0}, \operatorname{Arg}\left(z-z_{0}\right) \in\left(\theta_{0}, \theta_{1}\right) ?$$

Show that

$$\sinh \left(z^{-1}\right) \sim \frac{1}{2} \exp \left(z^{-1}\right) \quad \text { as } \quad z \rightarrow 0, \operatorname{Arg} z \in(-\pi / 2, \pi / 2),$$

and find the corresponding result in the sector $\operatorname{Arg} z \in(\pi / 2,3 \pi / 2)$.

What is meant by the asymptotic expansion

$$f(z) \sim \sum_{j=0}^{\infty} c_{j}\left(z-z_{0}\right)^{j} \quad \text { as } \quad z \rightarrow z_{0}, \operatorname{Arg}\left(z-z_{0}\right) \in\left(\theta_{0}, \theta_{1}\right) ?$$

Show that the coefficients $\left\{c_{j}\right\}_{j=0}^{\infty}$ are determined uniquely by $f$. Show that if $f$ is analytic at $z_{0}$, then its Taylor series is an asymptotic expansion for $f$ as $z \rightarrow z_{0}\left(\right.$ for any $\left.\operatorname{Arg}\left(z-z_{0}\right)\right)$.

Show that

$$u(x, t)=\int_{-\infty}^{\infty} \exp \left(-i k^{2} t+i k x\right) f(k) d k$$

defines a solution of the equation $i \partial_{t} u+\partial_{x}^{2} u=0$ for any smooth and rapidly decreasing function $f$. Use the method of stationary phase to calculate the leading-order behaviour of $u(\lambda t, t)$ as $t \rightarrow+\infty$, for fixed $\lambda$.