Quod Erat Demonstrandum

2014/08/11

吃驚積分

Filed under: Fun,Pure Mathematics — johnmayhk @ 9:18 下午
Tags:

\displaystyle \int_0^\infty \frac{\sin x}{x}dx=\frac{\pi}{2}

檢查一下:

http://www.wolframalpha.com/input/?i=integrate+sin%28x%29%2Fx+from+0+to+infinity

原來

\displaystyle \int_0^\infty \frac{\sin x}{x}\cdot \frac{\sin (x/3)}{(x/3)}dx=\frac{\pi}{2}

又檢查一下:

http://www.wolframalpha.com/input/?i=integrate+%28sin%28x%29%2Fx%29%28sin%28x%2F3%29%2F%28x%2F3%29%29+from+0+to+infinity

甚至

\displaystyle \int_0^\infty \frac{\sin x}{x}\cdot \frac{\sin (x/3)}{(x/3)}\cdot \frac{\sin (x/5)}{(x/5)}dx=\frac{\pi}{2}

又又檢查下:

http://www.wolframalpha.com/input/?i=integrate+%28sin%28x%29%2Fx%29%28sin%28x%2F3%29%2F%28x%2F3%29%29%28sin%28x%2F5%29%2F%28x%2F5%29%29+from+0+to+infinity

如此,我們有

\displaystyle \int_0^\infty \frac{\sin x}{x}\cdot \frac{\sin (x/3)}{(x/3)}\cdot \frac{\sin (x/5)}{(x/5)}\cdot \frac{\sin (x/7)}{(x/7)}dx=\frac{\pi}{2}

\displaystyle \int_0^\infty \frac{\sin x}{x}\cdot \frac{\sin (x/3)}{(x/3)}\cdot \frac{\sin (x/5)}{(x/5)}\cdot \frac{\sin (x/7)}{(x/7)}\cdot \frac{\sin (x/9)}{(x/9)}dx=\frac{\pi}{2}

\displaystyle \int_0^\infty \frac{\sin x}{x}\cdot \frac{\sin (x/3)}{(x/3)}\cdot \frac{\sin (x/5)}{(x/5)}\cdot \frac{\sin (x/7)}{(x/7)}\cdot \frac{\sin (x/9)}{(x/9)}\cdot \frac{\sin (x/11)}{(x/11)}dx=\frac{\pi}{2}

\displaystyle \int_0^\infty \frac{\sin x}{x}\cdot \frac{\sin (x/3)}{(x/3)}\cdot \frac{\sin (x/5)}{(x/5)}\cdot \frac{\sin (x/7)}{(x/7)}\cdot \frac{\sin (x/9)}{(x/9)}\cdot \frac{\sin (x/11)}{(x/11)}\cdot \frac{\sin (x/13)}{(x/13)}dx=\frac{\pi}{2}

你會猜想事情一直如常發展下去嗎?

小心,原來

\displaystyle \int_0^\infty \frac{\sin x}{x}\cdot \frac{\sin (x/3)}{(x/3)}\cdot \frac{\sin (x/5)}{(x/5)}\cdot \frac{\sin (x/7)}{(x/7)}\cdot \frac{\sin (x/9)}{(x/9)}\cdot \frac{\sin (x/11)}{(x/11)}\cdot \frac{\sin (x/13)}{(x/13)}\cdot \frac{\sin (x/15)}{(x/15)}dx=a\pi

其中

a=\frac{467807924713440738696537864469}{935615849440640907310521750000}

參考資料:

https://dl.dropboxusercontent.com/u/19150457/fea-borwein.pdf

P.512

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