Any advice you care to offer concerning the following exercise would be greatly appreciated.
Let $\phi (a) = \Gamma (a) \Gamma (1-a) \sin(\pi a)$.
Prove that $\phi$ extends to an infinitely differentiate function on the real line.
What I know and what I've tried: I have learned that $\Gamma$ extends to a smooth function on the complement of the non positive integers. In addition if we take for granted that $\phi (a) = \phi (a+1)$ and that $\phi (0) =\pi$ (here I really mean $\phi$'s extension) then all I need to show is that $\phi$ is smooth at $0$. I have tried various obvious strategies like computing the difference quotient and trying to compute the limit of $d^k\phi/da^k$ as $a$ approaches $0$ but I haven't been successful. I would be grateful for any help you care to offer. Thanks for your time.