すべての自然数の積(解析接続あり)

\begin{align*} \prod_{k=1}^{\infty}k & =\prod_{k=1}^{\infty}e^{\Log k}\\ & =\exp\left(\sum_{k=1}^{\infty}\Log k\right)\\ & =\exp\left(-\left[\sum_{k=1}^{\infty}-k^{-s}\Log k\right]_{s=0}\right)\\ & =\exp\left(-\left[\frac{d}{ds}\sum_{k=1}^{\infty}k^{-s}\right]_{s=0}\right)\\ & =\exp\left(-\left[\frac{d}{ds}\zeta\left(s\right)\right]_{s=0}\right)\\ & =\exp\left(-\zeta'\left(0\right)\right)\\ & =\exp\left(\Log\sqrt{2\pi}\right)\\ & =\sqrt{2\pi} \end{align*}