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> Show that there is a holomorphic function defined in the set $$\Omega=\lbrace z\in \mathbb{C}:\vert z \vert >4 \rbrace $$ whose derivative is $$\frac{z}{(z-1)(z-2)(z-3)}$$
>  Is there a holomorphic function on $\Omega$ whose derivative is $$\frac{z^2}{(z-1)(z-2)(z-3)}\,\,?$$
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I know that each holomorphic function on an open convex set has an antiderivative, but $\Omega$ isn't convex. Should I look for functions with these derivatives, such as $f(z)=\frac{1}{2}(\log(1-z)-4\log (2-z)+3\log (3-z))$ and $g(z)=\frac{1}{2}(9\log (z-3)-8\log (z-2)+\log(z-1))$, or are there any well-known propositions on finding antiderivatives in open, non-convex sets?