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)

\[ \prod_{k=1}^{\infty}\frac{\sqrt[2k-1]{\alpha}}{\sqrt[2k]{\alpha}}=2^{\Log\alpha} \]

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\(\psi\left(z\right)\)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*}

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分母と分子交互に根号の総乗

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