|
|
| Line 1: |
Line 1: |
| {{DISPLAYTITLE: List of formulae involving {{pi}} }}
| | Hello buddy. Let me introduce myself. I am Ron but I don't like when people use my full title. Delaware is our beginning location. One of my preferred hobbies is tenting and now I'm attempting to earn money with it. I am a cashier and I'll be promoted soon.<br><br>My page; [http://ghaziabadmart.com/oxwall/blogs/post/16946 ghaziabadmart.com] |
| {{Pi box}}
| |
| The following is a list of significant formulae involving the [[mathematical constant]] [[pi|{{pi}}]]. The list contains only formulae whose significance is established either in the article on the formula itself, the article on pi, or the one on numerical [[approximations of π|approximations of {{pi}}]].
| |
| | |
| ==Classical geometry==
| |
| :<math>C = 2 \pi r = \pi d\!</math>
| |
| | |
| where {{math|''C''}} is the circumference of a [[circle]], {{math|''r''}} is the radius and {{math|''d''}} is the diameter.
| |
| | |
| :<math>A = \pi r^2\!</math>
| |
| | |
| where {{math|A}} is the area of a circle and {{math|''r''}} is the radius.
| |
| | |
| :<math>V = {4 \over 3}\pi r^3\!</math>
| |
| | |
| where {{math|''V''}} is the volume of a [[sphere]] and {{math|''r''}} is the radius.
| |
| | |
| :<math>SA = 4\pi r^2\!</math>
| |
| | |
| where {{math|''SA''}} is the surface area of a sphere and {{math|''r''}} is the radius.
| |
| | |
| ==Analysis==
| |
| ===Integrals===
| |
| :<math>\int\limits_{-\infty}^{\infty} \text{sech}(x)dx = \pi \!</math>
| |
| | |
| <br />
| |
| | |
| <math>\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\!</math>
| |
| | |
| <br /> | |
| | |
| :<math>\int\limits_{-1}^1 \sqrt{1-x^2}\,dx = \frac{\pi}{2}\!</math>
| |
| | |
| <br /> | |
| | |
| :<math>\int\limits_{-1}^1\frac{dx}{\sqrt{1-x^2}} = \pi\!</math>
| |
| | |
| <br />
| |
| | |
| :<math>\int\limits_{-\infty}^\infty\frac{dx}{1+x^2} = \pi\!</math> (integral form of [[arctan]] over its entire domain, giving the period of [[Trigonometric_functions#tangent|tan]]).
| |
| | |
| <br />
| |
| | |
| :<math>\int\limits_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}\!</math> (see [[gaussian integral]]).
| |
| | |
| <br />
| |
| | |
| :<math>\oint\frac{dz}{z}=2\pi i\!</math> (when the path of integration winds once counterclockwise around 0. See also [[Cauchy's integral formula]])
| |
| | |
| <br />
| |
| | |
| :<math>\int\limits_{-\infty}^{\infty} \frac{\sin x}{x}\,dx=\pi \!</math>
| |
| | |
| <br />
| |
| | |
| :<math>\int\limits_0^1 {x^4(1-x)^4 \over 1+x^2}\,dx = {22 \over 7} - \pi\!</math> (see also [[Proof that 22/7 exceeds π]]).
| |
| | |
| <br />
| |
| | |
| ===Efficient infinite series===
| |
| :<math>\sum_{k=0}^\infty\frac{k!}{(2k+1)!!}=\sum_{k=0}^\infty\frac{2^k k!^2}{(2k+1)!}=\frac{\pi}{2}\!</math> (see also [[double factorial]])
| |
| | |
| <br />
| |
| | |
| :<math>12 \sum^\infty_{k=0} \frac{(-1)^k (6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 640320^{3k + 3/2}}=\frac{1}{\pi}\!</math> (see [[Chudnovsky algorithm]])
| |
| | |
| <br />
| |
| | |
| :<math>\frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}=\frac{1}{\pi}\!</math> (see [[Srinivasa Ramanujan#Mathematical achievements|Srinivasa Ramanujan]])
| |
| | |
| <br />
| |
| | |
| :<math>\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\!</math><ref>Cetin Hakimoglu-Brown [http://iamned.com/math/infiniteseries.pdf Derivation of Rapidly Converging Infinite Series]</ref>
| |
| | |
| <br />
| |
| | |
| The following are good for calculating arbitrary binary digits of {{pi}}:
| |
| :<math>\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\!</math> (see [[Bailey-Borwein-Plouffe formula]])
| |
| | |
| <br />
| |
| | |
| :<math>\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\!</math>
| |
| | |
| <br />
| |
| | |
| ===Other infinite series===
| |
| :<math>\zeta(2) = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots = \frac{\pi^2}{6}\!</math> (see also [[Basel problem]] and [[Riemann zeta function]])
| |
| | |
| <br />
| |
| | |
| :<math>\zeta(4)= \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \frac{1}{4^4} + \cdots = \frac{\pi^4}{90}\!</math>
| |
| | |
| <br />
| |
| | |
| :<math>\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)!}\!</math> , where ''B''<sub>2''n''</sub> is a [[Bernoulli number]].
| |
| | |
| <br />
| |
| | |
| :<math>\sum_{n=1}^{\infty} \frac{3^n - 1}{4^n}\, \zeta(n+1) = \pi\!</math><ref>[http://mathworld.wolfram.com/PiFormulas.html Weisstein, Eric W. "Pi Formulas", MathWorld]</ref>
| |
| | |
| <br />
| |
| | |
| :<math>\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}\!</math> (see [[Leibniz formula for pi]])
| |
| | |
| <br />
| |
| | |
| :<math>\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}\!</math>
| |
| | |
| <br />
| |
| | |
| :<math>\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}\!</math>
| |
| | |
| <br />
| |
| | |
| :<math>\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}\!</math>
| |
| | |
| <br />
| |
| | |
| :<math>\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}\!</math>
| |
| | |
| <br />
| |
| | |
| :<math>\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}\!</math>
| |
| <br />
| |
| | |
| :<math> \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 \! </math> ([[Leonhard Euler|Euler]])
| |
| :where the numerators are the odd primes; each denominator is the multiple of four nearest to the numerator.
| |
| | |
| <br />
| |
| | |
| :<math> \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 \!</math> (Euler, 1748)
| |
| | |
| : After the first two terms, the signs are determined as follows: If the denominator is a prime of the form 4''m'' - 1, the sign is positive; if the denominator is a prime of the form 4''m'' + 1, the sign is negative; for composite numbers, the sign is equal the product of the signs of its factors.<ref>[[Carl B. Boyer]], ''A History of Mathematics'', Chapter 21., p. 488-489</ref>
| |
| | |
| <br />
| |
| | |
| ===Machin-like formulae===
| |
| See also [[Machin-like formula]].
| |
| : <math>\frac{\pi}{4} = 4 \arctan\frac{1}{5} - \arctan\frac{1}{239} \!</math> (the original [[John Machin|Machin's]] formula)
| |
| <br />
| |
| :<math>\frac{\pi}{4} = \arctan 1</math>
| |
| <br />
| |
| :<math>\frac{\pi}{4} = \arctan\frac{1}{2} + \arctan\frac{1}{3}\!</math>
| |
| <br />
| |
| :<math>\frac{\pi}{4} = 2 \arctan\frac{1}{2} - \arctan\frac{1}{7}\!</math>
| |
| <br />
| |
| :<math>\frac{\pi}{4} = 2 \arctan\frac{1}{3} + \arctan\frac{1}{7}\!</math>
| |
| <br />
| |
| :<math>\frac{\pi}{4} = 5 \arctan\frac{1}{7} + 2 \arctan\frac{3}{79}\!</math>
| |
| <br />
| |
| :<math>\frac{\pi}{4} = 12 \arctan\frac{1}{49} + 32 \arctan\frac{1}{57} - 5 \arctan\frac{1}{239} + 12 \arctan\frac{1}{110443}\!</math>
| |
| <br />
| |
| :<math>\frac{\pi}{4} = 44 \arctan\frac{1}{57} + 7 \arctan\frac{1}{239} - 12 \arctan\frac{1}{682} + 24 \arctan\frac{1}{12943}\!</math>
| |
| <br />
| |
| | |
| ===Infinite series===
| |
| Some infinite series involving pi are:<ref>{{cite web |title=The world of Pi |url=http://www.pi314.net/eng/ramanujan.php |author= Simon Plouffe / David Bailey |publisher=Pi314.net |date= |accessdate=2011-01-29}}<br/>{{cite web|url=http://numbers.computation.free.fr/Constants/Pi/piSeries.html |title=Collection of series for {{pi}} |publisher=Numbers.computation.free.fr |date= |accessdate=2011-01-29}}</ref>
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| | <math>\pi=\frac{1}{Z}\!</math>|| <math>Z=\sum_{n=0}^{\infty } \frac{((2n)!)^3(42n+5)} {(n!)^6{16}^{3n+1}}\!</math>
| |
| |-
| |
| | <math>\pi=\frac{4}{Z}\!</math>|| <math>Z=\sum_{n=0}^{\infty } \frac{(-1)^n(4n)!(21460n+1123)} {(n!)^4{441}^{2n+1}{2}^{10n+1}}</math>
| |
| |-
| |
| | <math>\pi=\frac{4}{Z}\!</math>|| <math>Z=\sum_{n=0}^{\infty } \frac{(6n+1)\left ( \frac{1}{2} \right )^3_n} {{4^n}(n!)^3}\!</math>
| |
| |-
| |
| | <math>\pi=\frac{32}{Z}\!</math>|| <math>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}\!</math>
| |
| |-
| |
| | <math>\pi=\frac{27}{4Z}\!</math>|| <math>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}\!</math>
| |
| |-
| |
| | <math>\pi=\frac{15\sqrt{3}}{2Z}\!</math>|| <math>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}\!</math>
| |
| |-
| |
| | <math>\pi=\frac{85\sqrt{85}}{18\sqrt{3}Z}\!</math>|| <math>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}\!</math>
| |
| |-
| |
| | <math>\pi=\frac{5\sqrt{5}}{2\sqrt{3}Z} \!</math>|| <math>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}\!</math>
| |
| |-
| |
| | <math>\pi=\frac{2\sqrt{3}}{Z} \!</math>|| <math>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}}\!</math>
| |
| |-
| |
| | <math>\pi=\frac{\sqrt{3}}{9Z} \!</math>|| <math>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}}\!</math>
| |
| |-
| |
| | <math>\pi=\frac{2\sqrt{11}}{11Z} \!</math>|| <math>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}}\!</math>
| |
| |-
| |
| | <math>\pi=\frac{\sqrt{2}}{4Z} \!</math>|| <math>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}}\!</math>
| |
| |-
| |
| | <math>\pi=\frac{4\sqrt{5}}{5Z} \!</math>|| <math>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}}\!</math>
| |
| |-
| |
| | <math>\pi=\frac{4\sqrt{3}}{3Z} \!</math>|| <math>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}}\!</math>
| |
| |-
| |
| | <math> \pi=\frac{4}{Z}\!</math>|| <math>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}}\!</math>
| |
| |-
| |
| | <math>\pi=\frac{72}{Z} \!</math>|| <math>Z=\sum_{n=0}^{\infty } \frac{(-1)^n(4n)!(260n+23)}{(n!)^44^{4n}18^{2n}}\!</math>
| |
| |-
| |
| | <math>\pi=\frac{3528}{Z} \!</math>|| <math>Z=\sum_{n=0}^{\infty } \frac{(-1)^n(4n)!(21460n+1123)}{(n!)^44^{4n}882^{2n}}\!</math>
| |
| |}
| |
| | |
| where
| |
| | |
| : <math>(x)_n \!</math>
| |
| | |
| is the [[Pochhammer symbol]] for the falling factorial.
| |
| | |
| ===Infinite products===
| |
| :<math> \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} \!</math> (see also [[Wallis product]])
| |
| | |
| <br />
| |
| [[Franciscus Vieta|Vieta]]'s formula:
| |
| | |
| :<math>\frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdot \cdots = \frac2\pi\!</math>
| |
| | |
| === Three continued fractions ===
| |
| :<math>
| |
| \pi= {3 + \cfrac{1^2}{6 + \cfrac{3^2}{6 + \cfrac{5^2}{6 + \cfrac{7^2}{6 + \ddots\,}}}}}
| |
| </math>
| |
| <br />
| |
| :<math>
| |
| \pi = \cfrac{4}{1 + \cfrac{1^2}{3 + \cfrac{2^2}{5 + \cfrac{3^2}{7 + \cfrac{4^2}{9 + \ddots}}}}}
| |
| </math>
| |
| <br />
| |
| :<math>
| |
| \pi = \cfrac{4}{1 + \cfrac{1^2}{2 + \cfrac{3^2}{2 + \cfrac{5^2}{2 + \cfrac{7^2}{2 + \ddots}}}}}\,
| |
| </math>
| |
| | |
| For more on this third identity, see [[Euler's continued fraction formula]].
| |
| | |
| (See also [[continued fraction]] and [[generalized continued fraction]].)
| |
| | |
| ===Miscellaneous===
| |
| :<math>n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n\!</math> ([[Stirling's approximation]])
| |
| <br />
| |
| :<math>e^{i \pi} = -1\!</math> ([[Euler's identity]])
| |
| <br />
| |
| :<math>\sum_{k=1}^{n} \varphi (k) \sim \frac{3n^2}{\pi^2}\!</math> (see [[Euler's totient function]])
| |
| <br />
| |
| :<math>\sum_{k=1}^{n} \frac {\varphi (k)} {k} \sim \frac{6n}{\pi^2}\!</math> (see [[Euler's totient function]])
| |
| <br />
| |
| :<math>\Gamma\left({1 \over 2}\right)=\sqrt{\pi}\!</math> (see also [[gamma function]])
| |
| <br />
| |
| :<math>\pi = \frac{\Gamma\left({1/4}\right)^{4/3} \mathrm{agm}(1, \sqrt{2})^{2/3}}{2}\!</math> (where agm is the [[arithmetic-geometric mean]])
| |
| <br />
| |
| :<math>\lim_{n\rightarrow \infty}\frac{1}{n^2} \sum_{k=1}^n (n\;\bmod\;k) = 1-\frac{\pi^2}{12}\!</math> (where mod is the modulo function which gives the rest of a division this formula is getting better for higher n)
| |
| <br />
| |
| :<math> \pi = \lim_{n \rightarrow \infty} \frac{4}{n^2} \sum_{k=1}^n \sqrt{n^2 - k^2} </math> ([[Riemann sum]] to evaluate the area of the unit circle)
| |
| <br />
| |
| :<math> \pi = \lim_{n \rightarrow \infty} \frac{2^{4n}}{n {2n\choose n}^2} </math> (by [[Stirling's approximation]])
| |
| <br />
| |
| | |
| ==Physics==
| |
| *The [[cosmological constant]]:
| |
| ::<math>\Lambda = {{8\pi G} \over {3c^2}} \rho\!</math>
| |
| <br />
| |
| *[[Uncertainty principle|Heisenberg's uncertainty principle]]:
| |
| ::<math> \Delta x\, \Delta p \ge \frac{h}{4\pi} \!</math>
| |
| <br />
| |
| *[[Einstein's field equation]] of [[general relativity]]:
| |
| ::<math> R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R + \Lambda g_{\mu\nu} = {8 \pi G \over c^4} T_{\mu\nu} \!</math>
| |
| <br />
| |
| *[[Coulomb's law]] for the [[electric force]]:
| |
| ::<math> F = \frac{\left|q_1q_2\right|}{4 \pi \varepsilon_0 r^2}\!</math>
| |
| <br />
| |
| *[[Permeability (electromagnetism)|Magnetic permeability of free space]]:
| |
| ::<math> \mu_0 = 4 \pi \cdot 10^{-7}\,\mathrm{N/A^2}\!</math>
| |
| <br />
| |
| *Period of a simple [[pendulum]] with small amplitude
| |
| :::<math>T \approx 2\pi \sqrt\frac{L}{g}\!</math>
| |
| | |
| ==See also==
| |
| * [[Pi]]
| |
| * [[List of topics related to π]]
| |
| | |
| ==References==
| |
| {{reflist|2}}
| |
| | |
| ==Further reading==
| |
| * 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.
| |
| | |
| {{DEFAULTSORT:Formulae Involving Pi, List Of}}
| |
| [[Category:Pi]]
| |
| [[Category:Mathematics-related lists|Pi]]
| |
| [[Category:Pi algorithms]]
| |