沃利斯乘積，又称沃利斯公式，由數學家約翰·沃利斯在1655年时发现。
:
\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 = \frac{\pi}{2}.
:
也可以写为:
\frac{(2n)!!}{(2n-1)!!} \sim \sqrt{\pi n}\left(n \to \infty\right)
:
當時證明
今日多數的微積分教科書透過比較 \int_0^\pi \sin^nxdx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。
:\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
:
\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
:\begin{align}
\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{align}
嚴謹證明
先考慮不定積分 \int \sin^nxdx 有
\int \sin^nxdx
=- \int \sin^{n-1}xd\cos x
=- \cos x \sin^{n-1}x + \int \cos xd\sin^{n-1}x
=- \cos x \sin^{n-1}x + \int (n-1)\sin^{n-2}x\cos^2 xdx
=- \cos x \sin^{n-1}x + (n-1)\int \sin^{n-2}x(1 - \sin^2 x)dx
=- \cos x \sin^{n-1}x + (n-1)\int \sin^{n-2}xdx - (n-1)\int \sin^n x dx
故
\int \sin^nxdx =-\frac{1}{n} \cos x \sin^{n-1}x + \frac{n-1}{n}\int \sin^{n-2}xdx
\int_0^\frac{\pi}{2} \sin^nxdx = \frac{n-1}{n}\int_0^\frac{\pi}{2} \sin^{n-2}xdx
對整數m
\int_0^\frac{\pi}{2} \sin^{2m}xdx
= \frac{2m-1}{2m}\int_0^\frac{\pi}{2} \sin^{2m-2}xdx
= \frac{2m-1}{2m}\frac{2m-3}{2m-2}\int_0^\frac{\pi}{2} \sin^{2m-4}xdx
= ...
= \frac{2m-1}{2m}\frac{2m-3}{2m-2}...\frac{1}{2}\int_0^\frac{\pi}{2} \sin^{0}xdx
= \frac{2m-1}{2m}\frac{2m-3}{2m-2}...\frac{1}{2}\frac{\pi}{2}
另一方面
\int_0^\frac{\pi}{2} \sin^{2m+1}xdx
= \frac{2m}{2m+1}\int_0^\frac{\pi}{2} \sin^{2m-1}xdx
= \frac{2m}{2m+1}\frac{2m-2}{2m-1}\int_0^\frac{\pi}{2} \sin^{2m-3}xdx
= ...
= \frac{2m}{2m+1}\frac{2m-2}{2m-1}...\frac{2}{3}\int_0^\frac{\pi}{2} \sin xdx
= \frac{2m}{2m+1}\frac{2m-2}{2m-1}...\frac{2}{3}
兩式相除得
\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
\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
1 = \frac{\int_0^\frac{\pi}{2} \sin^{2m+1}xdx}{ \int_0^\frac{\pi}{2} \sin^{2m+1}xdx}
由夾擠定理知
\lim_{m \to \infty}1 = \lim_{m \to \infty} \frac{2m+1}{2m} = 1
故
\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
尋找 ζ(2)
我們可將上述的正弦乘積式化為泰勒级数：
: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
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