含圓周率的公式列表

下面是一個涉及數學常數π的公式列表。

古典幾何

$C=2\pi r=\pi d\!$

其中，$C$是一個圓的周長，$r$是半徑，$d$是直徑。

$A=\pi r^{2}\!$

其中$A$是一個圓的面積，$r$是半徑。

$V={4 \over 3}\pi r^{3}\!$

其中，$V$是一個球體的體積，$r$是半徑。

$A=4\pi r^{2}\!$

其中$A$是一個球體的表面積，$r$是半徑。

分析

積分

$\int \limits _{-\infty }^{\infty }{\text{sech}}(x)dx=\pi \!$

$\int _{0}^{\infty }{\frac {dx}{(x+1){\sqrt {x}}}}=\pi$

$\int \limits _{-1}^{1}{\sqrt {1-x^{2}}}\,dx={\frac {\pi }{2}}\!$

$\int \limits _{-1}^{1}{\frac {dx}{\sqrt {1-x^{2}}}}=\pi \!$

$\int \limits _{-\infty }^{\infty }{\frac {dx}{1+x^{2}}}=\pi \!$

$\int \limits _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}\!$ (參見 正態分佈)

$\oint {\frac {dz}{z}}=2\pi i\!$ (參見 柯西積分公式)

$\int \limits _{-\infty }^{\infty }{\frac {\sin(x)}{x}}\,dx=\pi \!$

$\int \limits _{0}^{1}{x^{4}(1-x)^{4} \over 1+x^{2}}\,dx={22 \over 7}-\pi \!$ (參見 證明22/7大於π)

高效的無窮級數

參見：Category:圓周率算法

${\frac {\pi }{2}}\!=\sum _{k=0}^{\infty }{\frac {k!}{(2k+1)!!}}=\sum _{k=0}^{\infty }{\frac {2^{k}k!^{2}}{(2k+1)!}}$ (參見 雙階乘)

${\frac {1}{\pi }}\!=12\sum _{k=0}^{\infty }{\frac {(-1)^{k}(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}640320^{3k+{\frac {3}{2}}}}}$ (參見 楚德諾夫斯基算法)

${\frac {1}{\pi }}\!={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}$ (參見拉馬努金)

$\pi \!={\frac {\sqrt {3}}{6^{5}}}\sum _{k=0}^{\infty }{\frac {[(4k)!]^{2}(6k)!}{9^{k+1}(12k)!(2k)!}}\left({\frac {127169}{12k+1}}-{\frac {1070}{12k+5}}-{\frac {131}{12k+7}}+{\frac {2}{12k+11}}\right)$

以下是任意位的二進制的π計算：:

$\pi \!=\sum _{k=0}^{\infty }{\frac {1}{16^{k}}}\left({\frac {4}{8k+1}}-{\frac {2}{8k+4}}-{\frac {1}{8k+5}}-{\frac {1}{8k+6}}\right)$ (參見 貝利－波爾溫－普勞夫公式)

$\pi ={\frac {1}{2^{6}}}\sum _{n=0}^{\infty }{\frac {{(-1)}^{n}}{2^{10n}}}\left(-{\frac {2^{5}}{4n+1}}-{\frac {1}{4n+3}}+{\frac {2^{8}}{10n+1}}-{\frac {2^{6}}{10n+3}}-{\frac {2^{2}}{10n+5}}-{\frac {2^{2}}{10n+7}}+{\frac {1}{10n+9}}\right)$

其他無窮級數

$\zeta (2)={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}\!$   (參見巴塞爾問題和黎曼ζ函數)

$\zeta (4)={\frac {1}{1^{4}}}+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{4^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}\!$

$\zeta (2n)={\frac {1}{1^{2n}}}+{\frac {1}{2^{2n}}}+{\frac {1}{3^{2n}}}+{\frac {1}{4^{2n}}}+\cdots =(-1)^{n+1}{\frac {B_{2n}(2\pi )^{2n}}{2(2n)!}}\!$

${\frac {\pi }{4}}\!=\sum _{n=0}^{\infty }{\left[{\frac {(-1)^{n}}{2n+1}}\right]}^{1}={\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots =\arctan {1}=\int _{0}^{1}{\frac {1}{1+x^{2}}}dx$   (參見Π的萊布尼茨公式)

${\frac {\pi ^{2}}{8}}\!=\sum _{n=0}^{\infty }{\left[{\frac {(-1)^{n}}{2n+1}}\right]}^{2}={\frac {1}{1^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}+\cdots$

${\frac {\pi ^{3}}{32}}\!=\sum _{n=0}^{\infty }{\left[{\frac {(-1)^{n}}{2n+1}}\right]}^{3}={\frac {1}{1^{3}}}-{\frac {1}{3^{3}}}+{\frac {1}{5^{3}}}-{\frac {1}{7^{3}}}+\cdots$

${\frac {\pi ^{4}}{96}}\!=\sum _{n=0}^{\infty }{\left[{\frac {(-1)^{n}}{2n+1}}\right]}^{4}={\frac {1}{1^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{5^{4}}}+{\frac {1}{7^{4}}}+\cdots$

${\frac {5\pi ^{5}}{1536}}\!=\sum _{n=0}^{\infty }{\left[{\frac {(-1)^{n}}{2n+1}}\right]}^{5}={\frac {1}{1^{5}}}-{\frac {1}{3^{5}}}+{\frac {1}{5^{5}}}-{\frac {1}{7^{5}}}+\cdots$

${\frac {\pi ^{6}}{960}}\!=\sum _{n=0}^{\infty }{\left[{\frac {(-1)^{n}}{2n+1}}\right]}^{6}={\frac {1}{1^{6}}}+{\frac {1}{3^{6}}}+{\frac {1}{5^{6}}}+{\frac {1}{7^{6}}}+\cdots$

${\frac {\pi }{4}}={\frac {3}{4}}\times {\frac {5}{4}}\times {\frac {7}{8}}\times {\frac {11}{12}}\times {\frac {13}{12}}\times {\frac {17}{16}}\times {\frac {19}{20}}\times {\frac {23}{24}}\times {\frac {29}{28}}\times {\frac {31}{32}}\times \cdots \!$ (歐拉)

$\pi ={1}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}-{\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}+{\frac {1}{9}}-{\frac {1}{10}}+{\frac {1}{11}}+{\frac {1}{12}}-{\frac {1}{13}}+\cdots \!$ (歐拉, 1748)

梅欽公式

參見梅欽公式.

${\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}\!$ (原始的梅欽公式.)

${\frac {\pi }{4}}=\arctan {\frac {1}{2}}+\arctan {\frac {1}{3}}\!$

${\frac {\pi }{4}}=2\arctan {\frac {1}{2}}-\arctan {\frac {1}{7}}\!$

${\frac {\pi }{4}}=2\arctan {\frac {1}{3}}+\arctan {\frac {1}{7}}\!$

${\frac {\pi }{4}}=5\arctan {\frac {1}{7}}+2\arctan {\frac {3}{79}}\!$

${\frac {\pi }{4}}=12\arctan {\frac {1}{49}}+32\arctan {\frac {1}{57}}-5\arctan {\frac {1}{239}}+12\arctan {\frac {1}{110443}}\!$

${\frac {\pi }{4}}=44\arctan {\frac {1}{57}}+7\arctan {\frac {1}{239}}-12\arctan {\frac {1}{682}}+24\arctan {\frac {1}{12943}}\!$

無窮級數

一些涉及圓周率的無窮級數:

$\pi ={\frac {1}{Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {[(2n)!]^{3}(42n+5)}{(n!)^{6}{16}^{3n+1}}}\!$
$\pi ={\frac {4}{Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(4n)!(21460n+1123)}{(n!)^{4}{441}^{2n+1}{2}^{10n+1}}}$
$\pi ={\frac {4}{Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(6n+1)\left({\frac {1}{2}}\right)_{n}^{3}}{{4^{n}}(n!)^{3}}}\!$
$\pi ={\frac {32}{Z}}\!$	$Z=\sum _{n=0}^{\infty }\left({\frac {{\sqrt {5}}-1}{2}}\right)^{8n}{\frac {(42n{\sqrt {5}}+30n+5{\sqrt {5}}-1)\left({\frac {1}{2}}\right)_{n}^{3}}{{64^{n}}(n!)^{3}}}\!$
$\pi ={\frac {27}{4Z}}\!$	$Z=\sum _{n=0}^{\infty }\left({\frac {2}{27}}\right)^{n}{\frac {(15n+2)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{3}}\right)_{n}\left({\frac {2}{3}}\right)_{n}}{(n!)^{3}}}\!$
$\pi ={\frac {15{\sqrt {3}}}{2Z}}\!$	$Z=\sum _{n=0}^{\infty }\left({\frac {4}{125}}\right)^{n}{\frac {(33n+4)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{3}}\right)_{n}\left({\frac {2}{3}}\right)_{n}}{(n!)^{3}}}\!$
$\pi ={\frac {85{\sqrt {85}}}{18{\sqrt {3}}Z}}\!$	$Z=\sum _{n=0}^{\infty }\left({\frac {4}{85}}\right)^{n}{\frac {(133n+8)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{6}}\right)_{n}\left({\frac {5}{6}}\right)_{n}}{(n!)^{3}}}\!$
$\pi ={\frac {5{\sqrt {5}}}{2{\sqrt {3}}Z}}\!$	$Z=\sum _{n=0}^{\infty }\left({\frac {4}{125}}\right)^{n}{\frac {(11n+1)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{6}}\right)_{n}\left({\frac {5}{6}}\right)_{n}}{(n!)^{3}}}\!$
$\pi ={\frac {2{\sqrt {3}}}{Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(8n+1)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{9}^{n}}}\!$
$\pi ={\frac {\sqrt {3}}{9Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(40n+3)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{49}^{2n+1}}}\!$
$\pi ={\frac {2{\sqrt {11}}}{11Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(280n+19)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{99}^{2n+1}}}\!$
$\pi ={\frac {\sqrt {2}}{4Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(10n+1)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{9}^{2n+1}}}\!$
$\pi ={\frac {4{\sqrt {5}}}{5Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(644n+41)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}5^{n}{72}^{2n+1}}}\!$
$\pi ={\frac {4{\sqrt {3}}}{3Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(28n+3)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{3^{n}}{4}^{n+1}}}\!$
$\pi ={\frac {4}{Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(20n+3)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{2}^{2n+1}}}\!$
$\pi ={\frac {72}{Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(4n)!(260n+23)}{(n!)^{4}4^{4n}18^{2n}}}\!$
$\pi ={\frac {3528}{Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(4n)!(21460n+1123)}{(n!)^{4}4^{4n}882^{2n}}}\!$

$(x)_{n}\!$是階乘冪中下降階乘冪的符號。

$\prod _{n=1}^{\infty }{\frac {4n^{2}}{4n^{2}-1}}={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdots ={\frac {4}{3}}\cdot {\frac {16}{15}}\cdot {\frac {36}{35}}\cdot {\frac {64}{63}}\cdots ={\frac {\pi }{2}}\!$ (參見沃利斯乘積)

弗朗索瓦·韋達的公式:

${\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdot \cdots ={\frac {2}{\pi }}\!$

連分數

$\pi ={3+{\cfrac {1^{2}}{6+{\cfrac {3^{2}}{6+{\cfrac {5^{2}}{6+{\cfrac {7^{2}}{6+\ddots \,}}}}}}}}}$

$\pi ={\cfrac {4}{1+{\cfrac {1^{2}}{3+{\cfrac {2^{2}}{5+{\cfrac {3^{2}}{7+{\cfrac {4^{2}}{9+\ddots }}}}}}}}}}$

$\pi ={\cfrac {4}{1+{\cfrac {1^{2}}{2+{\cfrac {3^{2}}{2+{\cfrac {5^{2}}{2+{\cfrac {7^{2}}{2+\ddots }}}}}}}}}}\,$

(參見連分數。)

雜項

$n!\approx {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\!$ (斯特靈公式)

$e^{i\pi }+1=0$ (歐拉恆等式)

$\sum _{k=1}^{n}\varphi (k)\approx {\frac {3n^{2}}{\pi ^{2}}}\!$

$\sum _{k=1}^{n}{\frac {\varphi (k)}{k}}\approx {\frac {6n}{\pi ^{2}}}\!$

$\Gamma \left({1 \over 2}\right)={\sqrt {\pi }}\!$ (伽瑪函數)

$\pi ={\frac {\Gamma \left({\frac {1}{4}}\right)^{\frac {4}{3}}\mathrm {agm} (1,{\sqrt {2}})^{\frac {2}{3}}}{2}}\!$

$\lim _{n\rightarrow \infty }{\frac {1}{n^{2}}}\sum _{k=1}^{n}(n\;{\bmod {\;}}k)=1-{\frac {\pi ^{2}}{12}}\!$

$\lim _{n\rightarrow \infty }10^{n+2}\cdot \sin \left({\frac {1^{\circ }}{\underbrace {55\cdots 55^{\circ }} _{\mathrm {n\;digits} }}}\right)=\pi \!$

$\lim _{n\rightarrow \infty }n\cdot \sin \left({\frac {180^{\circ }}{n}}\right)=\pi$

$\lim _{n\rightarrow \infty }{\frac {n}{\sqrt {2}}}\cdot {\sqrt {1-\cos \left({\frac {360^{\circ }}{n}}\right)}}=\pi$

物理

• 宇宙常數:

$\Lambda ={{8\pi G} \over {3c^{2}}}\rho \!$

• 不確定性原理:

$\Delta x\,\Delta p\geq {\frac {h}{4\pi }}\!$

• 愛因斯坦場方程:

$R_{ik}-{g_{ik}R \over 2}+\Lambda g_{ik}={8\pi G \over c^{4}}T_{ik}\!$

• 庫侖定律:

$F={\frac {\left|q_{1}q_{2}\right|}{4\pi \varepsilon _{0}r^{2}}}\!$

• 真空磁導率:

$\mu _{0}=4\pi \cdot 10^{-7}\,(\mathrm {N/A^{2}} )\!$

• 單擺周期

$T=2\pi {\sqrt {\frac {L}{g}}}\!$

拓展閱讀

• Peter Borwein, The Amazing Number Pi（頁面存檔備份，存於互聯網檔案館）
• Kazuya Kato, Nobushige Kurokawa, Saito Takeshi: Number Theory 1: Fermat's Dream. American Mathematical Society, Providence 1993, ISBN 0-8218-0863-X.

參見

• 圓周率