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:


 * $\forall x \in \R: \map f x = \log \, \size {2 \sin \pi x}$

Then $f$ is a replicative function.

Proof
We have that:

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

By Product Formula for Sine:

Hence the result.