偶数ゼータの通常型母関数

偶数ゼータの通常型母関数

\[ \sum_{k=1}^{\infty}\zeta(2k)x^{2k}=\frac{1}{2}\left(1-\pi x\tan^{-1}\left(\pi x\right)\right) \]

\begin{align*} \sum_{k=1}^{\infty}\zeta(2k)x^{2k} & =\sum_{k=1}^{\infty}\sum_{j=1}^{\infty}\frac{1}{j^{2k}}x^{2k}\\ & =\sum_{j=1}^{\infty}\sum_{k=1}^{\infty}\left(\frac{x}{j}\right)^{2k}\\ & =\sum_{j=1}^{\infty}\sum_{k=1}^{\infty}\left(\frac{x}{j}\right)^{2k}\\ & =\sum_{j=1}^{\infty}\left(\frac{x}{j}\right)^{2}\frac{1}{1-\left(\frac{x}{j}\right)^{2}}\\ & =x\sum_{j=1}^{\infty}\frac{x}{j^{2}-x^{2}}\\ & =-\frac{x}{2}\frac{d}{dx}\sum_{j=1}^{\infty}\log\left(j^{2}-x^{2}\right)\\ & =-\frac{x}{2}\frac{d}{dx}\log\left(\prod_{j=1}^{\infty}\left(j^{2}-x^{2}\right)\right)\\ & =-\frac{x}{2}\frac{d}{dx}\log\left(\frac{1}{\pi x}\left(\prod_{m=1}^{\infty}m^{2}\right)\pi x\left(\prod_{j=1}^{\infty}\frac{j^{2}-x^{2}}{j^{2}}\right)\right)\\ & =-\frac{x}{2}\frac{d}{dx}\log\left(\frac{1}{\pi x}\left(\prod_{m=1}^{\infty}m^{2}\right)\sin\left(\pi x\right)\right)\\ & =-\frac{x}{2}\frac{d}{dx}\left(\log\sin\left(\pi x\right)-\log x+\log\left(\frac{1}{\pi}\left(\prod_{m=1}^{\infty}m^{2}\right)\right)\right)\\ & =-\frac{x}{2}\left(\pi\tan^{-1}\left(\pi x\right)-\frac{1}{x}\right)\\ & =\frac{1}{2}\left(1-\pi x\tan^{-1}\left(\pi x\right)\right) \end{align*}

ページ情報

タイトル

偶数ゼータの通常型母関数

URL

https://www.nomuramath.com/ih5f369k/

SNSボタン