The equation

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

where $a$ is a constant with $\operatorname{Re} a>0$, has solutions of the form

$$w(z)=\int_{\gamma} e^{z t} f(t) d t$$

for suitably chosen contours $\gamma$ and some suitable function $f(t)$.

(a) Find $f(t)$ and determine the condition on $\gamma$, which you should express in terms of $z, t$ and $a$.

(b) Use the results of part (a) to show that $\gamma$ can be a finite contour and specify two possible finite contours with the help of a clearly labelled diagram. Hence, find the corresponding solution of the equation $(\dagger)$ in the case $a=1$.

(c) In the case $a=1$ and real $z$, show that $\gamma$ can be an infinite contour and specify two possible infinite contours with the help of a clearly labelled diagram. [Hint: Consider separately the cases $z>0$ and $z<0$.] Hence, find a second, linearly independent solution of the equation ( $\dagger$ ) in this case.