Show that the equation

$$z w^{\prime \prime}+2 k w^{\prime}+z w=0,$$

where $k$ is constant, has solutions of the form

$$w(z)=\int_{\gamma}\left(t^{2}+1\right)^{k-1} e^{z t} d t$$

provided that the path $\gamma$ is chosen so that $\left[\left(t^{2}+1\right)^{k} e^{z t}\right]_{\gamma}=0$.

(i) In the case Re $k>0$, show that there is a choice of $\gamma$ for which $w(0)=i B\left(k, \frac{1}{2}\right)$.

(ii) In the case $k=n / 2$, where $n$ is any integer, show that $\gamma$ can be a finite contour and that the corresponding solution satisfies $w(0)=0$ if $n \leqslant-1$.