By setting $w(z)=\int_{\gamma} f(t) e^{-z t} d t$, where $\gamma$ and $f(t)$ are to be suitably chosen, explain how to find integral representations of the solutions of the equation

$$z w^{\prime \prime}-k w=0$$

where $k$ is a non-zero real constant and $z$ is complex. Discuss $\gamma$ in the particular case that $z$ is restricted to be real and positive and distinguish the different cases that arise according to the $\operatorname{sign}$ of $k$.

Show that in this particular case, by choosing $\gamma$ as a closed contour around the origin, it is possible to express a solution in the form

$$w(z)=A \sum_{n=0}^{\infty} \frac{(z k)^{n+1}}{n !(n+1) !}$$

where $A$ is a constant.

Show also that for $k>0$ there are solutions that satisfy

$$w(z) \sim B z^{1 / 4} e^{-2 \sqrt{k z}} \quad \text { as } z \rightarrow \infty$$

where $B$ is a constant.