Logarithm of Absolute Value of 2 times Sine of pi x is Replicative Function

Theorem
Let $f: \R \to \R$ be the real function defined as:


 * $\forall x \in \R: f \left({x}\right) = \log \, \left \vert{2 \sin \pi x}\right\vert$

Then $f$ is a replicative function.

Proof
We have that:

Thus to demonstrate that $f$ is replicative, it is sufficient to demonstrate that:
 * $\displaystyle \prod_{k \mathop = 0}^{n - 1} \left({2 \sin \pi \left({x + \frac k n}\right)}\right) = 2 \sin \pi n x$

We have: