ウォリス積分の拡張2重階乗表示
ウォリス積分の拡張2重階乗表示
\[ \int_{0}^{\frac{\pi}{2}}\sin^{n}\theta d\theta=\frac{\left(n-1\right)!^{2}}{\left(n\right)!^{2}}\sqrt{\frac{\pi}{2}} \]
\[ \int_{0}^{\frac{\pi}{2}}\sin^{n}\theta d\theta=\frac{\left(n-1\right)!^{2}}{\left(n\right)!^{2}}\sqrt{\frac{\pi}{2}} \]
*
\(x!^{n}\)は拡張多重階乗。\begin{align*}
\int_{0}^{\frac{\pi}{2}}\sin^{n}\theta d\theta & =\frac{1}{2}B\left(\frac{n+1}{2},\frac{1}{2}\right)\\
& =\frac{1}{2}\frac{\Gamma\left(\frac{n+1}{2}\right)\Gamma\left(\frac{1}{2}\right)}{\Gamma\left(\frac{n+2}{2}\right)}\\
& =\frac{\Gamma\left(\frac{1}{2}\right)}{2}\frac{\left(n-1\right)!^{2}2^{-\frac{n-2}{2}}\frac{1}{2}!}{\left(n\right)!^{2}2^{-\frac{n-1}{2}}\frac{1}{2}!}\cmt{x!^{n}=n^{\frac{x-1}{n}}\frac{\left(\frac{x}{n}\right)!}{\left(\frac{1}{n}\right)!}}\\
& =\frac{\left(n-1\right)!^{2}}{\left(n\right)!^{2}}\sqrt{\frac{\pi}{2}}
\end{align*}
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| タイトル | ウォリス積分の拡張2重階乗表示 |
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多重階乗と拡張多重階乗の定義
\[
\left(x\right)!^{n}=n^{\frac{x-1}{n}}\frac{\left(\frac{x}{n}\right)!}{\left(\frac{1}{n}\right)!}
\]
多重階乗同士の関係
\[
\left(qn+r\right)!^{n}=r!^{n}\frac{\left(qn+r\right)!_{n}}{r!_{n}}
\]
(拡張)多重階乗の逆数和
\[
\sum_{k=0}^{n}\frac{1}{\left(ak+b\right)!_{a}}=\frac{e^{\frac{1}{a}}a^{\frac{b}{a}}\Gamma\left(\frac{b}{a}+1\right)}{b!_{a}}\left(\frac{\Gamma\left(n+\frac{b}{a}+1,\frac{1}{a}\right)}{\Gamma\left(n+\frac{b}{a}+1\right)}-\frac{\Gamma\left(\frac{b}{a},\frac{1}{a}\right)}{\Gamma\left(\frac{b}{a}\right)}\right)
\]
(拡張)多重階乗と階乗の関係
\[
\left(an+b\right)!_{a}=\frac{a^{n}b!_{a}\left(n+\frac{b}{a}\right)!}{\left(\frac{b}{a}\right)!}
\]