I have the following equation:
holds for all complex number $s$.
I am little confused about this case: If $f(s)≠0$ then I can deduce directely that $g(s)=1$ as the unique case. However, when $f(s)=0$ for one and isolated $s$ (root) then $g(s)$ would be any function. This is the point. Is this case have some relations with the analytic continuation principle.