(a) Show that there does not exist a holomorphic function $f$ on $\mathbb C \backslash \{1, -1\}$ so that $$f'(z) = \frac{1}{(z^2-1)^{2019}} \text{ for all } z \in \mathbb C \backslash \{1, -1\}.$$ (b) Show that there exist a set $L \subset \mathbb C$ and a holomorphic function $F$ on $\mathbb C \backslash L$ so that $L$ has Hausdorff dimension $1$, and $$F'(z) = \frac{1}{(z^2-1)^{2019}} \text{ for all } z \in \mathbb C \backslash L.$$
I didn't manage to figure out (a), I believe (a) should solve after integrate along some curve to arrive in contradiction, but I don't know a good way of changing variables in $z$ to make $z^2-1$ a simple variable. Also I don't know much properties about Hausdorff dimension, but what properties of a Hausdorff dimension $1$ have that makes the holomorphic function exist?