The 3x+1 map is give as
$$f(x) = \begin{cases} \frac{3x+1}{2} & \text{ if x odd} \\ \frac{x}{2} & \text{ else} \end{cases}$$ with domain $\mathbb{N}.$
On this wikipedia article, I found that this function can be extended to a smooth function on $\mathbb{C}$ to
$$f(z)=\frac{1}{4}(1 + 4z - (1 + 2z)\cos(\pi z)).$$
Is this extension unique or is there any reason to consider this particular extension?
