I am trying to find a method to solve this problem. I am given this condition for a complex function $f$:
$$f(e^{-n})=\bigg(e^{n}+\frac{1}{e^n}\bigg)\sin(e^{-n})$$
and I am asked to find all functions that verify this property and are holomorphic in the complex plane.
I have tried to use the result that states that, if $A$ is a subset of the complex plane with a limit point, and $f=0$ in $A$, then $f=0$ in the complex plane. But I am unsure of which function to take to apply this result.