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:

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

We have that:

$\ds \sum_{k \mathop = 0}^{n - 1} \map f {x + \frac k n}$

$=$

$\ds \sum_{k \mathop = 0}^{n - 1} \log \, \size {2 \sin \pi \paren {x + \frac k n} }$

$\ds $

$=$

$\ds \log \prod_{k \mathop = 0}^{n - 1} \size {2 \sin \pi \paren {x + \frac k n} }$

Thus to demonstrate that $f$ is replicative, it is sufficient to demonstrate that:

$\ds \prod_{k \mathop = 0}^{n - 1} \paren {2 \sin \pi \paren {x + \frac k n} } = 2 \sin \pi n x$

$\ds 2 \sin \pi n x$

$=$

$\ds 2^n \prod_{k \mathop = 0}^{n - 1} \paren {\map \sin {\pi x + \frac {k \pi} n} }$

$\ds $

$=$

$\ds \prod_{k \mathop = 0}^{n - 1} \paren {2 \sin \pi \paren {x + \frac k n} }$

Hence the result.

$\blacksquare$