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Classical geometry C = 2 \pi r = \pi d\! where C is the circumference of a circle, r is the radius and d is the diameter. A = \pi r^2\! where A is the area of a circle and r is the radius. V = {4 \over 3}\pi r^3\! where V is the volume of a sphere and r is the radius. SA = 4\pi r^2\! where SA is the surface area of a sphere and r is the radius. Analysis Integrals \int\limits_{-\infty}^{\infty} \text{sech}(x)dx = \pi \! \int\limits_{-\infty}^{\infty} \int\limits_{t}^{\infty} e^{-1/2t^2-x^2+xt} dxdt = \int\limits_{-\infty}^{\infty} \int\limits_{t}^{\infty} e^{^-t^2-1/2x^2+xt} dxdt = \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\! (integral form of arctan over its entire domain, giving the period of tan). \int\limits_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}\! (see gaussian integral). \oint\frac{dz}{z}=2\pi i\! (when the path of integration winds once counterclockwise around 0. See also Cauchy's integral formula) \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\! (see also Proof that 22/7 exceeds π). Efficient infinite series \sum_{k=0}^\infty\frac{k!}{(2k+1)!!}=\sum_{k=0}^\infty\frac{2^k k!^2}{(2k+1)!}=\frac{\pi}{2}\! (see also double factorial) 12 \sum^\infty_{k=0} \frac{(-1)^k (6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 640320^{3k + 3/2}}=\frac{1}{\pi}\! (see Chudnovsky algorithm) \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}=\frac{1}{\pi}\! (see Srinivasa Ramanujan) \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)=\pi\![1] The following are good for calculating arbitrary binary digits of π: \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)=\pi\! (see Bailey-Borwein-Plouffe formula) \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)=\pi\! Other infinite series \zeta(2) = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots = \frac{\pi^2}{6}\! (see also Basel problem and Riemann zeta function) \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) = \sum_{k=1}^{\infty} \frac{1}{k^{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)!}\! , where B2n is a Bernoulli number. \sum_{n=1}^{\infty} \frac{3^n - 1}{4^n}\, \zeta(n+1) = \pi\![2] \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} = \frac{\pi}{4}\! (see Leibniz formula for pi) \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^2}{8}\! \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^3}{32}\! \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{\pi^4}{96}\! \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{5\pi^5}{1536}\! \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^6}{960}\! \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 \! (Euler) where the numerators are the odd primes; each denominator is the multiple of four nearest to the numerator. \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 \! (Euler, 1748) After the first two terms, the signs are determined as follows: If the denominator is a prime of the form 4m - 1, the sign is positive; if the denominator is a prime of the form 4m + 1, the sign is negative; for composite numbers, the sign is equal the product of the signs of its factors.[3] Machin-like formulae See also Machin-like formula. \frac{\pi}{4} = 4 \arctan\frac{1}{5} - \arctan\frac{1}{239} \! (the original Machin's formula) \frac{\pi}{4} = \arctan 1 \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}\! Infinite series Some infinite series involving pi are:[4] \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 )^3_n} {{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 )^3_n} {{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!)^35^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!)^44^{4n}18^{2n}}\! \pi=\frac{3528}{Z} \! Z=\sum_{n=0}^{\infty } \frac{(-1)^n(4n)!(21460n+1123)}{(n!)^44^{4n}882^{2n}}\! where (x)_n \! is the Pochhammer symbol for the falling factorial. Infinite products \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} \! (see also Wallis product) Vieta's formula: \frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdot \cdots = \frac2\pi\! Three continued fractions \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}}}}}\, For more on this third identity, see Euler's continued fraction formula. (See also continued fraction and generalized continued fraction.) Miscellaneous n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n\! (Stirling's approximation) e^{i \pi} = -1\! (Euler's identity) \sum_{k=1}^{n} \varphi (k) \sim \frac{3n^2}{\pi^2}\! (see Euler's totient function) \sum_{k=1}^{n} \frac {\varphi (k)} {k} \sim \frac{6n}{\pi^2}\! (see Euler's totient function) \Gamma\left({1 \over 2}\right)=\sqrt{\pi}\! (see also gamma function) \pi = \frac{\Gamma\left({1/4}\right)^{4/3} \mathrm{agm}(1, \sqrt{2})^{2/3}}{2}\! (where agm is the arithmetic-geometric mean) \lim_{n\rightarrow \infty}\frac{1}{n^2} \sum_{k=1}^n (n\;\bmod\;k) = 1-\frac{\pi^2}{12}\! (where mod is the modulo function which gives the rest of a division this formula is getting better for higher n) \pi = \lim_{n \rightarrow \infty} \frac{4}{n^2} \sum_{k=1}^n \sqrt{n^2 - k^2} (Riemann sum to evaluate the area of the unit circle) \pi = \lim_{n \rightarrow \infty} \frac{2^{4n}}{n {2n\choose n}^2} (by Stirling's approximation) Physics The cosmological constant: \Lambda = {{8\pi G} \over {3c^2}} \rho\! Heisenberg's uncertainty principle: \Delta x\, \Delta p \ge \frac{h}{4\pi} \! Einstein's field equation of general relativity: R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R + \Lambda g_{\mu\nu} = {8 \pi G \over c^4} T_{\mu\nu} \! Coulomb's law for the electric force: F = \frac{\left|q_1q_2\right|}{4 \pi \varepsilon_0 r^2}\! Magnetic permeability of free space: \mu_0 = 4 \pi \cdot 10^{-7}\,\mathrm{N/A^2}\! Period of a simple pendulum with small amplitude T \approx 2\pi \sqrt\frac{L}{g}\! |
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