Meaning of Euler Reflection Formula

Question

A book that I am reading says that Euler reflection formula $$\Gamma (x)\Gamma (1-x)=\frac{\pi}{\sin \pi x}$$, in a sense, shows that $1/\Gamma(x)$ is half of the sine function.

Does anyone understand what the book is talking about?

Answer 1

I think the book meant analytic structure. The right side has simple poles at $x\in\mathbb{Z}$, $\Gamma(x)$ at $x\in\mathbb{Z}_{\le 0}$ and $\Gamma(1-x)$ at $x\in\mathbb{Z}_{>0}$. So to prove the identity, it suffices to check coincidence of the residues and behaviour at infinity.

One may also think about comparing product representations: \begin{align} \frac{1}{\Gamma(x)}&=x\prod_{n=1}^{\infty}\frac{1+\frac{x}{n}}{(1+\frac{1}{n})^x},\\ \sin\pi x &= \pi x \prod_{n=1}^{\infty}\left(1+\frac{x}{n}\right)\left(1-\frac{x}{n}\right). \end{align}

This picture has further generalizations - for example, $q$-deformed gamma function is in the same sense a "half" of the Jacobi theta function.