含圆周率的公式列表

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 常態分布)

 

　
: $\oint {\frac {dz}{z}}=2\pi i\!$ (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

　
: ${\frac {\pi }{2}}\!=\sum _{k=0}^{\infty }{\frac {k!}{(2k+1)!!}}=\sum _{k=0}^{\infty }{\frac {2^{k}k!^{2}}{(2k+1)!}}$ (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位的 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:

　
: $(x)_{n}\!$ 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

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

## 參考來源

## 拓展閱讀

* Peter Borwein, _The Amazing Number Pi（頁面存檔備份，存於網際網路檔案館）_
* 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