$(a)$ Show that there is a complex analytic function defined on the set $U=\{z\in\mathbb{C} \ | \ |z|>4\}$ whose derivative is $\displaystyle\frac{z}{(z-1)(z-2)(z-3)}$.
$(b)$ Is there a complex analytic function on $U$ whose derivative is $\displaystyle\frac{z^2}{(z-1)(z-2)(z-3)}$?
The problem clearly reduces to find a analytic function $f(z)$ on $U$ such that $f'(z)=\displaystyle\frac{z}{(z-1)(z-2)(z-3)}$. But if I integrate both sides over the contour of $U$, the RHS is analytic on $U$. So $f(z)=0$. But this result is too absurd to consider. Again same goes for $(b)$, but which is not the case since we have to prove no such $f$ would exist. I considered the CR-equations but the given function is too large to handle for this. What is the best approach for this problem? Any help is appreciated.
