数学中有什么难以置信的结论？

发布时间：

2023-08-24 12:48

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最近在一本书上看到了这么一个结论：

$\int_{0}^{\infty}\cos x \cdot\frac{\sin4x}{x}dx=\frac{\pi}{2},$

$\int_{0}^{\infty}\cos x \cdot\cos \frac{x}{2}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2},$

$\int_{0}^{\infty}\cos x \cdot\cos \frac{x}{2}\cdot\cos \frac{x}{3}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2},$

$\int_{0}^{\infty}\cos x \cdot\cos \frac{x}{2}\cdot\cos \frac{x}{3}\cdot\cos \frac{x}{4}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2},$

$\int_{0}^{\infty}\cos x \cdot\cos \frac{x}{2}\cdot\cos \frac{x}{3}\cdot\cos \frac{x}{4}\cdot\cos \frac{x}{5}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2},$

$\int_{0}^{\infty}\cos x \cdot\cos \frac{x}{2}\cdot\cos \frac{x}{3}\cdot\cos \frac{x}{4}\cdot\cos \frac{x}{5}\cdot\cos \frac{x}{6}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2}.$

这也太巧了吧？其实还没完！

$\int_{0}^{\infty}\cos x \cdot\cos \frac{x}{2}\cdot\cos \frac{x}{3}\cdot\cdot\cdot\cos \frac{x}{6}\cdot\cos \frac{x}{7}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2},$

$\int_{0}^{\infty}\cos x \cdot\cos \frac{x}{2}\cdot\cos \frac{x}{3}\cdot\cdot\cdot\cos \frac{x}{7}\cdot\cos \frac{x}{8}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2},$

$\int_{0}^{\infty}\cos x \cdot\cos \frac{x}{2}\cdot\cos \frac{x}{3}\cdot\cdot\cdot\cos \frac{x}{8}\cdot\cos \frac{x}{9}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2},$

$\int_{0}^{\infty}\cos x \cdot\cos \frac{x}{2}\cdot\cos \frac{x}{3}\cdot\cdot\cdot\cos \frac{x}{9}\cdot\cos \frac{x}{10}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2},$

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\cdot\cdot\cdot\cdot\cdot\cdot$

$\int_{0}^{\infty}\cos x \cdot\cos \frac{x}{2}\cdot\cos \frac{x}{3}\cdot\cdot\cdot\cos \frac{x}{18}\cdot\cos \frac{x}{19}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2},$

$\int_{0}^{\infty}\cos x \cdot\cos \frac{x}{2}\cdot\cos \frac{x}{3}\cdot\cdot\cdot\cos \frac{x}{19}\cdot\cos \frac{x}{20}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2}.$

此时有一种令人难以置信的感觉，但还没完！

$\int_{0}^{\infty}\cos x \cdot\cos \frac{x}{2}\cdot\cos \frac{x}{3}\cdot\cdot\cdot\cos \frac{x}{20}\cdot\cos \frac{x}{21}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2},$

$\int_{0}^{\infty}\cos x \cdot\cos \frac{x}{2}\cdot\cos \frac{x}{3}\cdot\cdot\cdot\cos \frac{x}{21}\cdot\cos \frac{x}{22}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2},$

$\int_{0}^{\infty}\cos x \cdot\cos \frac{x}{2}\cdot\cos \frac{x}{3}\cdot\cdot\cdot\cos \frac{x}{22}\cdot\cos \frac{x}{23}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2},$

$\int_{0}^{\infty}\cos x \cdot\cos \frac{x}{2}\cdot\cos \frac{x}{3}\cdot\cdot\cdot\cos \frac{x}{23}\cdot\cos \frac{x}{24}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2},$

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\cdot\cdot\cdot\cdot\cdot\cdot$

$\int_{0}^{\infty}\cos x \cdot\cos \frac{x}{2}\cdot\cos \frac{x}{3}\cdot\cdot\cdot\cos \frac{x}{28}\cdot\cos \frac{x}{29}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2},$

$\int_{0}^{\infty}\cos x \cdot\cos \frac{x}{2}\cdot\cos \frac{x}{3}\cdot\cdot\cdot\cos \frac{x}{29}\cdot\cos \frac{x}{30}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2}.$

看到这里我想大家应该会猜测：

猜想：对任意正整数 $n,$ 有

$~~~~~~\int_{0}^{\infty}\left( \prod_{k=1}^{n}\cos\frac{x}{k} \right)\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2}.$

但是，但是，当 $n=31$ 时，

$\int_{0}^{\infty}\left( \prod_{k=1}^{31}\cos\frac{x}{k} \right)\cdot\frac{\sin4x}{x}dx=1.57079632533<\frac{\pi}{2}.$

误差大约为 $0.00000000146,$ 猜想戛然而止！

其实上面这些神奇的现象源于下述结论：

定理：设 $a_{0},a_{1},\cdot\cdot\cdot,a_{m},b_{1},\cdot\cdot\cdot ,b_{n}$ 为实数，且满足下列条件

$~~~~~~a_{0}>0,~~~a_{0}\geq\sum_{i=1}^{m}{|a_{i}|}+\sum_{j=1}^{n}{|b_{j}|}.$

则我们有

$\int_{0}^{\infty}\left( \prod_{i=0}^{m}\frac{\sin a_{i}x}{x} \right)\cdot\left( \prod_{j=1}^{n}\cos b_{j}x \right)dx=\left( \prod_{i=1}^{m}{ a_{i}}\right)\cdot\frac{\pi}{2}.$

上面的例子其实是取 $a_{0}=4$ . 当 $n\leq30$ 时， $\sum_{k=1}^{n}{\frac{1}{k}}\leq4$ ，满足条件，从而有

$~~~~~~\int_{0}^{\infty}\left( \prod_{k=1}^{n}\cos\frac{x}{k} \right)\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2}.$

当 $n\geq31$ 时， $\sum_{k=1}^{n}{\frac{1}{k}}>4$ ，结论不再成立.

而如果我们取 $a_{0}=4,~a_{1}=1$ . 则会有

$\int_{0}^{\infty}\cos x \cdot\frac{\sin x}{x}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2},$

$\int_{0}^{\infty}\cos x\cdot\cos \frac{x}{2} \cdot\frac{\sin x}{x}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2},$

$\int_{0}^{\infty}\cos x\cdot\cos \frac{x}{2}\cdot\cos \frac{x}{3} \cdot\frac{\sin x}{x}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2},$

$\int_{0}^{\infty}\cos x\cdot\cos \frac{x}{2}\cdot\cos \frac{x}{3} \cdot\cos \frac{x}{4}\cdot\frac{\sin x}{x}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2},$

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\cdot\cdot\cdot\cdot\cdot\cdot$

$\int_{0}^{\infty}\cos x \cdot\cos \frac{x}{2}\cdot\cos \frac{x}{3}\cdot\cdot\cdot\cos \frac{x}{8}\cdot\cos \frac{x}{9}\cdot\frac{\sin x}{x}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2},$

$\int_{0}^{\infty}\cos x \cdot\cos \frac{x}{2}\cdot\cos \frac{x}{3}\cdot\cdot\cdot\cos \frac{x}{9}\cdot\cos \frac{x}{10}\cdot\frac{\sin x}{x}\cdot\frac{\sin4x}{x}dx=\frac{\pi}{2},$

但是

$\int_{0}^{\infty}\cos x \cdot\cos \frac{x}{2}\cdot\cos \frac{x}{3}\cdot\cdot\cdot\cos \frac{x}{10}\cdot\cos \frac{x}{11}\cdot\frac{\sin x}{x}\cdot\frac{\sin4x}{x}dx\ne\frac{\pi}{2}.$