沃利斯乘积

今日多數的微積分教科書透過比較${\displaystyle \int _{0}^{\pi }\sin ^{n}xdx}$在n是奇數或是偶數，甚至是接近無窮大的情況下，發現即使將n增加一就會發生不一樣的情形。在那時，微積分尚未存在，而且有關數學收斂的分析工具也還未俱全，所以完成這證明較現今有相當的難度。從現在來看，從欧拉公式中的正弦展開式得到此乘積是必然的結果。

${\displaystyle {\frac {\sin(x)}{x}}=\left(1-{\frac {x^{2}}{\pi ^{2}}}\right)\left(1-{\frac {x^{2}}{4\pi ^{2}}}\right)\left(1-{\frac {x^{2}}{9\pi ^{2}}}\right)\cdots =\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{n^{2}\pi ^{2}}}\right),}$

在x = π/2時

${\displaystyle {\frac {2}{\pi }}=\prod _{n=1}^{\infty }\left(1-{\frac {1}{4n^{2}}}\right)=\left(1-{\frac {1}{2^{2}}}\right)\left(1-{\frac {1}{2^{2}\cdot 4}}\right)\left(1-{\frac {1}{2^{2}\cdot 9}}\right)\cdots }$

${\displaystyle {\begin{aligned}{\frac {\pi }{2}}&{}=\prod _{n=1}^{\infty }\left({\frac {4n^{2}}{4n^{2}-1}}\right)\\&{}=\prod _{n=1}^{\infty }{\frac {(2n)(2n)}{(2n-1)(2n+1)}}={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdots \end{aligned}}}$

先考慮不定積分${\displaystyle \int \sin ^{n}xdx}$有

${\displaystyle \int \sin ^{n}xdx}$

${\displaystyle =-\int \sin ^{n-1}xd\cos x}$

${\displaystyle =-\cos x\sin ^{n-1}x+\int \cos xd\sin ^{n-1}x}$

${\displaystyle =-\cos x\sin ^{n-1}x+\int (n-1)\sin ^{n-2}x\cos ^{2}xdx}$

${\displaystyle =-\cos x\sin ^{n-1}x+(n-1)\int \sin ^{n-2}x(1-\sin ^{2}x)dx}$

${\displaystyle =-\cos x\sin ^{n-1}x+(n-1)\int \sin ^{n-2}xdx-(n-1)\int \sin ^{n}xdx}$

故

${\displaystyle \int \sin ^{n}xdx=-{\frac {1}{n}}\cos x\sin ^{n-1}x+{\frac {n-1}{n}}\int \sin ^{n-2}xdx}$

${\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{n}xdx={\frac {n-1}{n}}\int _{0}^{\frac {\pi }{2}}\sin ^{n-2}xdx}$

對整數m

${\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{2m}xdx}$

${\displaystyle ={\frac {2m-1}{2m}}\int _{0}^{\frac {\pi }{2}}\sin ^{2m-2}xdx}$

${\displaystyle ={\frac {2m-1}{2m}}{\frac {2m-3}{2m-2}}\int _{0}^{\frac {\pi }{2}}\sin ^{2m-4}xdx}$

${\displaystyle =...}$

${\displaystyle ={\frac {2m-1}{2m}}{\frac {2m-3}{2m-2}}...{\frac {1}{2}}\int _{0}^{\frac {\pi }{2}}\sin ^{0}xdx}$

${\displaystyle ={\frac {2m-1}{2m}}{\frac {2m-3}{2m-2}}...{\frac {1}{2}}{\frac {\pi }{2}}}$

另一方面

${\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{2m+1}xdx}$

${\displaystyle ={\frac {2m}{2m+1}}\int _{0}^{\frac {\pi }{2}}\sin ^{2m-1}xdx}$

${\displaystyle ={\frac {2m}{2m+1}}{\frac {2m-2}{2m-1}}\int _{0}^{\frac {\pi }{2}}\sin ^{2m-3}xdx}$

${\displaystyle =...}$

${\displaystyle ={\frac {2m}{2m+1}}{\frac {2m-2}{2m-1}}...{\frac {2}{3}}\int _{0}^{\frac {\pi }{2}}\sin xdx}$

${\displaystyle ={\frac {2m}{2m+1}}{\frac {2m-2}{2m-1}}...{\frac {2}{3}}}$

兩式相除得

${\displaystyle {\frac {\int _{0}^{\frac {\pi }{2}}\sin ^{2m}xdx}{\int _{0}^{\frac {\pi }{2}}\sin ^{2m+1}xdx}}={\frac {{\frac {2m-1}{2m}}{\frac {2m-3}{2m-2}}...{\frac {1}{2}}{\frac {\pi }{2}}}{{\frac {2m}{2m+1}}{\frac {2m-2}{2m-1}}...{\frac {2}{3}}}}}$

故

${\displaystyle {\frac {\pi }{2}}={\frac {2}{1}}{\frac {2}{3}}{\frac {4}{3}}{\frac {4}{5}}...{\frac {2m}{2m-1}}{\frac {2m}{2m+1}}{\frac {\int _{0}^{\frac {\pi }{2}}\sin ^{2m}xdx}{\int _{0}^{\frac {\pi }{2}}\sin ^{2m+1}xdx}}={\frac {\int _{0}^{\frac {\pi }{2}}\sin ^{2m}xdx}{\int _{0}^{\frac {\pi }{2}}\sin ^{2m+1}xdx}}\prod _{n=1}^{m}{\frac {2n}{2n-1}}\cdot {\frac {2n}{2n+1}}}$

又因為

${\displaystyle 1={\frac {\int _{0}^{\frac {\pi }{2}}\sin ^{2m+1}xdx}{\int _{0}^{\frac {\pi }{2}}\sin ^{2m+1}xdx}}<{\frac {\int _{0}^{\frac {\pi }{2}}\sin ^{2m}xdx}{\int _{0}^{\frac {\pi }{2}}\sin ^{2m+1}xdx}}<{\frac {\int _{0}^{\frac {\pi }{2}}\sin ^{2m-1}xdx}{\int _{0}^{\frac {\pi }{2}}\sin ^{2m+1}xdx}}={\frac {2m+1}{2m}}}$

由夾擠定理知

${\displaystyle \lim _{m\to \infty }1=\lim _{m\to \infty }{\frac {2m+1}{2m}}=1}$

故

${\displaystyle {\frac {\pi }{2}}=\prod _{n=1}^{\infty }{\frac {2n}{2n-1}}\cdot {\frac {2n}{2n+1}}={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdots }$

我們可將上述的正弦乘積式化為泰勒级数：

${\displaystyle x\left(1-{\frac {x^{2}}{\pi ^{2}}}\right)\left(1-{\frac {x^{2}}{4\pi ^{2}}}\right)\left(1-{\frac {x^{2}}{9\pi ^{2}}}\right)\cdots =x-{\frac {1}{3!}}x^{3}+{\frac {1}{5!}}x^{5}-\cdots }$