I've recently learned that the cotangent satisfies the following functional equation:
(true for $f(z)\neq 0$).
Can we solve this equation for real or complex functions $f?$ Can we give additional conditions such that $\cot$ is the only real or complex function satisfying these conditions and the equation? Or is there perhaps a different functional equation better suited for this purpose?
I'm asking this because I know about such a characterization of the real function $\exp$.
Please note that I know very little about functional equations. I've only seen two examples dealt with in my courses.