| 圓周率 |
| --- |
|  |
| **3.1415926535897932384626433...** |
| 
運用 |
| * 圓的面積
* 周長
* 含圆周率的公式列表
 |
| 
證明 |
| * 無理性
* 超越性
 |
| 
值 |
| * 約率（證明22/7大於π）
* 密率
* 近似值
* 割圓術
 |
| 
人物 |
| * 阿基米德
* 刘徽
* 祖冲之
* 玛达瓦（英语：）
* 威廉·琼斯
* 約翰·梅欽
* 约翰·伦奇
* 鲁道夫·范科伊伦
* 阿耶波多
 |
| 
歷史 |
| * 年表
 |
| 
文化 |
| * 節日
 |
| 
相關主題 |
| * 化圓為方
* 巴塞爾問題
* 连续的六个9
 |
| * 
* 论
* 编
 |

下面是一个涉及数学常数π的公式列表。

古典几何
----

C=2\pi r=\pi d\!
其中，C是一个圆的周长，r是半径，d是直径。

A=\pi r^{2}\!
其中A是一个圆的面积，r是半径。

V={4 \over 3}\pi r^{3}\!
其中，V是一个球体的体积，r是半径。

A=4\pi r^{2}\!
其中A是一个球体的表面积，r是半径。

分析
--

### 积分

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

{\displaystyle \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 }}\! (参见 正态分布)
  

\oint {\frac {dz}{z}}=2\pi i\! (参见 柯西积分公式)
  

\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 \! (参见 證明22/7大於π)
### 高效的无穷级数

参见：Category:圆周率算法
{\frac {\pi }{2}}\!=\sum \_{{k=0}}^{\infty }{\frac {k!}{(2k+1)!!}}=\sum \_{{k=0}}^{\infty }{\frac {2^{k}k!^{2}}{(2k+1)!}} (参见 双阶乘)
  

{\displaystyle {\frac {1}{\pi }}\!=12\sum \_{k=0}^{\infty }{\frac {(-1)^{k}(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}640320^{3k+{\frac {3}{2}}}}}} (参见 楚德诺夫斯基算法)
  

{\frac {1}{\pi }}\!={\frac {2{\sqrt {2}}}{9801}}\sum \_{{k=0}}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{{4k}}}} (参见拉马努金)
  

\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)
  

以下是任意位的二进制的π计算：:

\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) (参见 贝利－波尔温－普劳夫公式)
  

\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)
  

### 其他无穷级数

\zeta (2)={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}\!   (参见巴塞尔问题和黎曼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)!}}\!
  

{\displaystyle {\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}   (参见Π的莱布尼茨公式)
  

{\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 \! (欧拉)
  

\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 \! (欧拉, 1748)
  

### 梅钦公式

参见梅钦公式.

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

{\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}}\!
### 无穷级数

一些涉及圆周率的无穷级数:

|  |  |
| --- | --- |
| \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}\!是阶乘幂中下降阶乘幂的符号。
\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}}\! (参见沃利斯乘积)
弗朗索瓦·韦达的公式:

{\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 }\!
### 连分数

\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 }}}}}}}}}}\,
(参见连分数。)

### 杂项

n!\approx {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\! (斯特灵公式)
  

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 }}\! (伽玛函数)
  

\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}}\!
  

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

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

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

物理
--

* 宇宙常数:

\Lambda ={{8\pi G} \over {3c^{2}}}\rho \!
* 不确定性原理:

\Delta x\,\Delta p\geq {\frac {h}{4\pi }}\!
* 爱因斯坦场方程:

R\_{{ik}}-{g\_{{ik}}R \over 2}+\Lambda g\_{{ik}}={8\pi G \over c^{4}}T\_{{ik}}\!
* 库仑定律:

F={\frac {\left|q\_{1}q\_{2}\right|}{4\pi \varepsilon \_{0}r^{2}}}\!
* 真空磁导率:

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

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

参考来源
----

1. ↑  Cetin Hakimoglu-Brown Derivation of Rapidly Converging Infinite Series （页面存档备份，存于）
2. ↑  Carl B. Boyer, *A History of Mathematics*, Chapter 21.
3. ↑  Simon Plouffe / David Bailey.. Pi314.net.  [2011-01-29]. （原始内容存档于2013-06-23）.   
. Numbers.computation.free.fr.  [2011-01-29]. （原始内容存档于2013-06-23）.

拓展阅读
----

* Peter Borwein, *The Amazing Number Pi（页面存档备份，存于）*
* Kazuya Kato, Nobushige Kurokawa, Saito Takeshi: *Number Theory 1: Fermat's Dream.* American Mathematical Society, Providence 1993, ISBN 0-8218-0863-X.

参见
--

* 圓周率