Can we give a definition of the cotangent based on a functional equation?

Question

I've recently learned that the cotangent satisfies the following functional equation:

$$\dfrac1{f(z)}=f(z)-2f(2z)$$

(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.

Answer 1

This might be related. The Herglotz trick is essentially the statement that $\pi\cot(\pi z)$ is the unique meromorphic function $f(z)$ satisfying:

$f(z)$ is defined for $z\in\mathbb{C}\backslash\mathbb{Z}$

$f(z+1)=f(z)$

$f(-z)=f(z)$

$-f(z+\frac{1}{2})=f(z)-2f(2z)$

$\lim_{z\to0}\left(f(z)-\frac{1}{z}\right)=0$