Finding an example of an analytic function, 0 on set of points (1/n) As per the title, I want a function $f(z)$ which is analytic on $\mathbb{C}$ on the set of points $\{1/n\}$ for $n \in \mathbb{Z}^+$ and with $f(z) \neq 0$.  What would this look like if the function just had to be analytic on $\mathbb{C} - {0}$?
$f(z)=\sin ({\pi\over z})$ may serve
$f(z)=\sin ({\pi\over z})$ may serve