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This is a homework problem from section 5.2.3 in Ahlfors that I have been struggling on for a long time. We want to show the canonical product representation $$\sin \pi(z+\alpha)=e^{\pi z \cot(\pi \alpha)}\prod_{n=-\infty}^\infty \left(1+\frac{z}{n+\alpha}\right)e^{-z/(n+\alpha)}$$ whenever $\alpha$ is not an integer. A hint is to let the factor at the front of the product be $g(z)$ and find $g'(z)/g(z)$.

I see that $\sin \pi(z+\alpha)$ has simple zeroes at $z=n-\alpha$ for $n\in \mathbb{Z}$, and we can use the Weierstraß factorization theorem, but I do not really know how to use the hint, as well as why we have $+\frac{z}{n+\alpha}$ instead of $-\frac{z}{n-\alpha}$. Maybe I'm just missing something here-- would anyone have any pointers?

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