Analytic function with assigned zeros

Question

Is there an example of an analytic function in the unit disc whose zeros are only the points $z_n=1-1/n$?

Answer 1

Another one: $$ f(z) = \frac{1}{\Gamma\left(\frac{1}{z-1}\right)} $$ This is analytic in the plane, except one point $z=1$, and has zeros exactly $1-1/n$, $n=1,2,3\dots$. Unlike Henning's, which also has zeros ${} \gt 1$.

Answer 2

How about $f(z) = \sin(\frac{\pi}{1-z})$?

Answer 3

For the general question of functions with prescribed zeros, consider Weierstraß' factorization theorem.