ウォリス積分の値

ウォリス積分の値

\(m\in\mathbb{N}_{0}\)とする。

(1)

\[ \int_{0}^{\frac{\pi}{2}}\sin^{2m}\theta d\theta=\frac{C(2m,m)}{4^{m}}\frac{\pi}{2} \]

(2)

\[ \int_{0}^{\frac{\pi}{2}}\sin^{2m+1}\theta d\theta=\frac{4^{m}}{(2m+1)C(2m,m)} \]

(0)

\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)} \end{align*}

(1)

\(n=2m(m\in\mathbb{N}_{0})\)のとき、
\begin{align*} \int_{0}^{\frac{\pi}{2}}\sin^{2m}\theta d\theta & =\frac{1}{2}\frac{\Gamma\left(m+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\right)}{\Gamma\left(m+1\right)}\\ & =\frac{1}{2}\frac{(2m-1)!\sqrt{\pi}\sqrt{\pi}}{2^{2m-1}(m-1)!m!}\\ & =\frac{C(2m,m)}{4^{m}}\frac{\pi}{2} \end{align*}

(2)

\(n=2m+1(m\in\mathbb{N}_{0})\)のとき、
\begin{align*} \int_{0}^{\frac{\pi}{2}}\sin^{2m+1}\theta d\theta & =\frac{1}{2}\frac{\Gamma\left(m+1\right)\Gamma\left(\frac{1}{2}\right)}{\Gamma\left(m+\frac{3}{2}\right)}\\ & =\frac{1}{2}\frac{(m+1)!\sqrt{\pi}2^{2m+1}m!}{(2m+1)!\sqrt{\pi}}\\ & =\frac{2^{2m}(m!)^{2}}{(2m+1)!}\\ & =\frac{4^{m}}{(2m+1)C(2m,m)} \end{align*}

(0)-2

\begin{align*} \int_{0}^{\frac{\pi}{2}}\sin^{n}\theta d\theta & =\int_{0}^{\frac{\pi}{2}}\sin^{n-1}\theta\sin\theta d\theta\\ & =-\left[\sin^{n-1}\theta\cos\theta\right]_{0}^{\frac{\pi}{2}}+(n-1)\int_{0}^{\frac{\pi}{2}}\sin^{n-2}\theta\cos^{2}\theta d\theta\\ & =(n-1)\int_{0}^{\frac{\pi}{2}}\left(\sin^{n-2}\theta-\sin^{n}\theta\right)d\theta\\ & =\frac{n-1}{n}\int_{0}^{\frac{\pi}{2}}\sin^{n-2}\theta d\theta \end{align*}

これより、

(1)-2

\(n=2m(m\in\mathbb{N}_{0})\)のとき、
\begin{align*} \int_{0}^{\frac{\pi}{2}}\sin^{2m}\theta d\theta & =\int_{0}^{\frac{\pi}{2}}\sin^{0}\theta d\theta\prod_{k=0}^{m-1}\frac{\int_{0}^{\frac{\pi}{2}}\sin^{2k+2}\theta d\theta}{\int_{0}^{\frac{\pi}{2}}\sin^{2k}\theta d\theta}\\ & =\frac{\pi}{2}\prod_{k=0}^{m-1}\frac{2k+1}{2k+2}\\ & =\frac{\pi}{2}\frac{(2m-1)!!}{(2m)!!}\\ & =\frac{\pi}{2}\frac{(2m-1)!}{2^{2m-1}m!(m-1)!}\\ & =\frac{C(2m,m)}{4^{m}}\frac{\pi}{2} \end{align*}

(2)-2

\(n=2m+1(m\in\mathbb{N}_{0})\)のとき、
\begin{align*} \int_{0}^{\frac{\pi}{2}}\sin^{2m+1}\theta d\theta & =\int_{0}^{\frac{\pi}{2}}\sin\theta d\theta\prod_{k=0}^{m-1}\frac{\int_{0}^{\frac{\pi}{2}}\sin^{2k+3}\theta d\theta}{\int_{0}^{\frac{\pi}{2}}\sin^{2k+1}\theta d\theta}\\ & =\prod_{k=0}^{m-1}\frac{2k+2}{2k+3}\\ & =\frac{(2m)!!}{(2m+1)!!}\\ & =\frac{2^{2m}(m!)^{2}}{(2m+1)!}\\ & =\frac{4^{m}}{(2m+1)C(2m,m)} \end{align*}

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ウォリス積分の値

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