Show that the equation

$$z w^{\prime \prime}-(1+z) w^{\prime}+2(1-z) w=0$$

has solutions of the form $w(z)=\int_{\gamma} e^{z t} f(t) d t$, where

$$f(t)=\frac{1}{(t-2)(t+1)^{2}}$$

provided that $\gamma$ is suitably chosen.

Hence find the general solution, evaluating the integrals explicitly. Show that the general solution is entire, but that there is no solution that satisfies $w(0)=0$ and $w^{\prime}(0) \neq 0$.