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 \!
:\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
:\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)
:\frac{1}{\pi}\!=12 \sum^\infty_{k=0} \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^\infty_{k=0} \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)!}\!
: \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 = + \frac + \frac + \frac - \frac + \frac + \frac + \frac + \frac - \frac + \frac + \frac - \frac + \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:
: (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{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdot \cdots = \frac2\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}\!
:\lim_{n\rightarrow \infty} 10^{n+2}\cdot \sin\left(\frac{1^\circ}{\underbrace{55\cdots55^\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
*宇宙常数:
::\Lambda = {{8\pi G} \over {3c^2}} \rho\!
*不确定性原理:
:: \Delta x\, \Delta p \ge \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_1q_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
::T = 2\pi \sqrt\frac{L}{g}\!
参考来源
拓展阅读
- Peter Borwein, [http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P159.pdf The Amazing Number Pi]
- 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
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