The equation

$$z w^{\prime \prime}+w=0$$

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 required condition on $\gamma$, which you should express in terms of $z$ and $t$.

(b) Use the result of part (a) to specify a possible contour with the help of a clearly labelled diagram.