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今回はボールウェイン積分の紹介です。
ボールウェイン積分とは
\(\displaystyle\int_{0}^{\infty} \displaystyle\frac{\sin x}{x}\biggl(\prod_{k=1}^{n}\displaystyle\frac{\sin a_{k} x}{a_{k} x}\biggr)dx=\displaystyle\frac{\pi}{2}\)
ただし、\(\displaystyle\sum_{k=1}^{n} |a_{k}|\leq 1\)
有名な例
この式は以下の例があるので有名になっています。
\(\displaystyle\int_{0}^{\infty} \displaystyle\frac{\sin x}{x}dx=\displaystyle\frac{\pi}{2}\) \(n=0\)
\(\displaystyle\int_{0}^{\infty} \displaystyle\frac{\sin x}{x}\cdot \displaystyle\frac{\sin \frac{x}{3}}{\frac{x}{3}}dx=\displaystyle\frac{\pi}{2}\) \(a_{1}=\displaystyle\frac{1}{3}\)、\(a_{2}=\displaystyle\frac{1}{5}\)
\(\displaystyle\int_{0}^{\infty} \displaystyle\frac{\sin x}{x}\cdot \displaystyle\frac{\sin \frac{x}{3}}{\frac{x}{3}}\cdot \displaystyle\frac{\sin \frac{x}{5}}{\frac{x}{5}}dx=\displaystyle\frac{\pi}{2}\) \(a_{1}=\displaystyle\frac{1}{3}\)、\(a_{2}=\displaystyle\frac{1}{5}\)、\(a_{3}=\displaystyle\frac{1}{7}\)
これを続けていくと次まで続く。 \(\cdots\)
\(\displaystyle\int_{0}^{\infty} \displaystyle\frac{\sin x}{x}\cdot \displaystyle\frac{\sin \frac{x}{3}}{\frac{x}{3}}\cdot \displaystyle\frac{\sin \frac{x}{5}}{\frac{x}{5}}\cdot \displaystyle\frac{\sin \frac{x}{7}}{\frac{x}{7}}\cdot \displaystyle\frac{\sin \frac{x}{9}}{\frac{x}{9}}\cdot \displaystyle\frac{\sin \frac{x}{11}}{\frac{x}{11}}\cdot \displaystyle\frac{\sin \frac{x}{13}}{\frac{x}{13}}dx=\displaystyle\frac{\pi}{2}\)
※\(\displaystyle\frac{1}{3}+\displaystyle\frac{1}{5}+\displaystyle\frac{1}{7}+\displaystyle\frac{1}{9}+\displaystyle\frac{1}{11}+\displaystyle\frac{1}{13}\leq 1\)
しかし、この次突然崩れる。
\(\displaystyle\int_{0}^{\infty} \displaystyle\frac{\sin x}{x}\cdot \displaystyle\frac{\sin \frac{x}{3}}{\frac{x}{3}}\cdot \displaystyle\frac{\sin \frac{x}{5}}{\frac{x}{5}}\cdot \displaystyle\frac{\sin \frac{x}{7}}{\frac{x}{7}}\cdot \displaystyle\frac{\sin \frac{x}{9}}{\frac{x}{9}}\cdot \displaystyle\frac{\sin \frac{x}{11}}{\frac{x}{11}}\cdot \displaystyle\frac{\sin \frac{x}{13}}{\frac{x}{13}}\cdot \displaystyle\frac{\sin \frac{x}{15}}{\frac{x}{15}}dx\)
\(=\displaystyle\frac{467807924713440738696537864469}{935615849440640907310521750000}\pi \)
※数値はWikipedia参照。
※\(\displaystyle\frac{1}{3}+\displaystyle\frac{1}{5}+\displaystyle\frac{1}{7}+\displaystyle\frac{1}{9}+\displaystyle\frac{1}{11}+\displaystyle\frac{1}{13}+\displaystyle\frac{1}{15}\geq 1\)
