分母と分子交互に根号の総乗
分母と分子交互に根号の総乗
(1)
\[ \prod_{k=1}^{n}\frac{\sqrt[2k-1]{\alpha}}{\sqrt[2k]{\alpha}}=\pow\left(\alpha,\frac{1}{2}\left(\psi\left(n+\frac{1}{2}\right)-\psi\left(n+1\right)+2\log2\right)\right) \](2)
\[ \prod_{k=1}^{\infty}\frac{\sqrt[2k-1]{\alpha}}{\sqrt[2k]{\alpha}}=2^{\Log\alpha} \]-
\(\psi\left(z\right)\)はディガンマ関数(1)
\begin{align*} \prod_{k=1}^{n}\frac{\sqrt[2k-1]{\alpha}}{\sqrt[2k]{\alpha}} & =\prod_{k=1}^{n}\frac{\alpha^{\frac{1}{2k-1}}}{\alpha^{\frac{1}{2k}}}\\ & =\prod_{k=1}^{n}\alpha^{\frac{1}{2k-1}-\frac{1}{2k}}\\ & =\pow\left(\alpha,\sum_{k=1}^{n}\left(\frac{1}{2k-1}-\frac{1}{2k}\right)\right)\\ & =\pow\left(\alpha,\sum_{k=1}^{n}\frac{1}{2k-1}-\frac{1}{2}\sum_{k=1}^{n}\frac{1}{k}\right)\\ & =\pow\left(\alpha,\frac{1}{2}\left\{ \psi\left(n-\frac{1}{2}+1\right)-\psi\left(1-\frac{1}{2}\right)\right\} -\frac{1}{2}\left\{ \psi\left(n+1\right)-\psi\left(1\right)\right\} \right)\\ & =\pow\left(\alpha,\frac{1}{2}\left\{ \psi\left(n+\frac{1}{2}\right)-\psi\left(n+1\right)+\psi\left(1\right)-\psi\left(\frac{1}{2}\right)\right\} \right)\\ & =\pow\left(\alpha,\frac{1}{2}\left\{ \psi\left(n+\frac{1}{2}\right)-\psi\left(n+1\right)-\gamma-\left(-\gamma-2\Log2\right)\right\} \right)\\ & =\pow\left(\alpha,\frac{1}{2}\left(\psi\left(n+\frac{1}{2}\right)-\psi\left(n+1\right)+2\log2\right)\right) \end{align*}(2)
\begin{align*} \prod_{k=1}^{\infty}\frac{\sqrt[2k-1]{\alpha}}{\sqrt[2k]{\alpha}} & =\prod_{k=1}^{\infty}\frac{\alpha^{\frac{1}{2k-1}}}{\alpha^{\frac{1}{2k}}}\\ & =\prod_{k=1}^{\infty}\alpha^{\frac{1}{2k-1}-\frac{1}{2k}}\\ & =\pow\left(\alpha,\sum_{k=1}^{\infty}\left(\frac{-\left(-1\right)^{2k-1}}{2k-1}-\frac{\left(-1\right)^{2k}}{2k}\right)\right)\\ & =\pow\left(\alpha,-\sum_{k=1}^{\infty}\frac{\left(-1\right)^{k}}{k}\right)\\ & =\pow\left(\alpha,\Log\left(1+1\right)\right)\\ & =\alpha^{\Log2}\\ & =2^{\Log\alpha} \end{align*}ページ情報
| タイトル | 分母と分子交互に根号の総乗 |
| URL | https://www.nomuramath.com/wqy0jjoy/ |
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総和・総乗・積分の順序・区間反転公式
\[
\sum_{k=a}^{b}f\left(k\right)=\sum_{k=a}^{b}f\left(a+b-k\right)
\]
1のn乗根のべき乗の総和
\[
\sum_{k=0}^{n-1}\left(\omega_{n}^{\;k}\right)^{m}=n\delta_{0,\mod\left(m,n\right)}
\]
1-1+1-1+…と続く総和
\[
\sum_{k=1}^{n}\left(-1\right)^{k+1}=\frac{1}{2}+\frac{\left(-1\right)^{n+1}}{2}
\]
積の形の無限多重根号
\[
\sqrt[a_{1}]{r_{1}\sqrt[a_{2}]{r_{2}\cdots\sqrt[a_{n}]{r_{n}}}}=\exp\left\{ \sum_{k=1}^{n}\left(\Log\left(r_{k}\right)\prod_{j=1}^{k}\frac{1}{a_{j}}\right)\right\}
\]