Zerrenda:Π duten formulak

Hona hemen π konstante matematikoarekin zerikusia duten formulen zerrenda.

Geometria klasikoa

$\pi =L/d\!$

non L d diametroko zirkunferentzia baten luzera baita.

$A=\pi r^{2}\!$

non A r erradioko zirkulu baten azalera baita.

$V={4 \over 3}\pi r^{3}\!$

non V baita r erradioko esfera baten bolumena.

$S=4\pi r^{2}\!$

non S r erradioko esfera baten kanpo-azalera baita.

Fisikoa

• Konstante kosmologikoa:

$\Lambda ={{8\pi G} \over {3c^{2}}}\rho \!$

• Heisenbergen ziurgabetasun-printzipioa:

$\Delta x\,\Delta p\geq {\frac {h}{4\pi }}\!$

• Erlatibitate orokorraren Einsteinen eremu-ekuazioak:

$R_{\mu \nu }-{\frac {1}{2}}g_{\mu \nu }R+\Lambda g_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu }\!$

• Indar elektrikoaren Culomb-en legea:

$F={\frac {\left|q_{1}q_{2}\right|}{4\pi \varepsilon _{0}r^{2}}}\!$

• Iragazkortasun magnetikoa hutsean:

$\mu _{0}=4\pi \cdot 10^{-7}\,\mathrm {N/A^{2}} \!$

• Anplitude txikiko pendulu sinple baten periodoa:

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

• Gilborduraren formula:

$F={\frac {\pi ^{2}EI}{L^{2}}}$

Identitateak

Integralak

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

$\int \limits _{-\infty }^{\infty }\int \limits _{t}^{\infty }e^{-1/2t^{2}-x^{2}+xt}dxdt=\int \limits _{-\infty }^{\infty }\int \limits _{t}^{\infty }e^{^{-}t^{2}-1/2x^{2}+xt}dxdt=\pi \!$

$\int \limits _{-1}^{1}{\sqrt {1-x^{2}}}\,dx={\frac {\pi }{2}}\!$

$\int \limits _{-1}^{1}{\frac {dx}{\sqrt {1-x^{2}}}}=\pi \!$

$\int \limits _{-\infty }^{\infty }{\frac {dx}{1+x^{2}}}=\pi \!$ (arku tangente baten forma integrala bere eremu osoan).

$\int \limits _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}\!$ (ikus Gauß-en Integrala).

$\oint {\frac {dz}{z}}=2\pi i\!$ (Ikus, halaber, Cauchyren formula integral)

$\int \limits _{-\infty }^{\infty }{\frac {\sin x}{x}}\,dx=\pi \!$

$\int \limits _{0}^{1}{x^{4}(1-x)^{4} \over 1+x^{2}}\,dx={22 \over 7}-\pi \!$

Serie infinitu eraginkorrak

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

$12\sum _{k=0}^{\infty }{\frac {(-1)^{k}(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}640320^{3k+3/2}}}={\frac {1}{\pi }}\!$

${\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}={\frac {1}{\pi }}\!$ (ikus Srinivasa Ramanujan)

${\frac {\sqrt {3}}{6^{5}}}\sum _{k=0}^{\infty }{\frac {((4k)!)^{2}(6k)!}{9^{k+1}(12k)!(2k)!}}\left({\frac {127169}{12k+1}}-{\frac {1070}{12k+5}}-{\frac {131}{12k+7}}+{\frac {2}{12k+11}}\right)=\pi \!$

Identitate hauek baliagarriak dira π-ren digitu bitar arbitrarioak kalkulatzeko:

$\sum _{k=0}^{\infty }{\frac {1}{16^{k}}}\left({\frac {4}{8k+1}}-{\frac {2}{8k+4}}-{\frac {1}{8k+5}}-{\frac {1}{8k+6}}\right)=\pi \!$

${\frac {1}{2^{6}}}\sum _{n=0}^{\infty }{\frac {{(-1)}^{n}}{2^{10n}}}\left(-{\frac {2^{5}}{4n+1}}-{\frac {1}{4n+3}}+{\frac {2^{8}}{10n+1}}-{\frac {2^{6}}{10n+3}}-{\frac {2^{2}}{10n+5}}-{\frac {2^{2}}{10n+7}}+{\frac {1}{10n+9}}\right)=\pi \!$

Beste serie infinitu batzuk

$\zeta (2)={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}\!$ (ikus, halaber, Basileako arazoa eta Riemann-en zeta funtzioa)

$\zeta (4)={\frac {1}{1^{4}}}+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{4^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}\!$

$\zeta (2n)=\sum _{k=1}^{\infty }{\frac {1}{k^{2n}}}\,={\frac {1}{1^{2n}}}+{\frac {1}{2^{2n}}}+{\frac {1}{3^{2n}}}+{\frac {1}{4^{2n}}}+\cdots =(-1)^{n+1}{\frac {B_{2n}(2\pi )^{2n}}{2(2n)!}}\!$ non B2n Bernoulliren zenbaki bat baita.

$\sum _{n=1}^{\infty }{\frac {3^{n}-1}{4^{n}}}\,\zeta (n+1)=\pi \!$

$\sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{1}={\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots =\arctan {1}={\frac {\pi }{4}}\!$ (Leibnizko seriea)

$\sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{2}={\frac {1}{1^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}+\cdots ={\frac {\pi ^{2}}{8}}\!$

$\sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{3}={\frac {1}{1^{3}}}-{\frac {1}{3^{3}}}+{\frac {1}{5^{3}}}-{\frac {1}{7^{3}}}+\cdots ={\frac {\pi ^{3}}{32}}\!$

$\sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{4}={\frac {1}{1^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{5^{4}}}+{\frac {1}{7^{4}}}+\cdots ={\frac {\pi ^{4}}{96}}\!$

$\sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{5}={\frac {1}{1^{5}}}-{\frac {1}{3^{5}}}+{\frac {1}{5^{5}}}-{\frac {1}{7^{5}}}+\cdots ={\frac {5\pi ^{5}}{1536}}\!$

$\sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{6}={\frac {1}{1^{6}}}+{\frac {1}{3^{6}}}+{\frac {1}{5^{6}}}+{\frac {1}{7^{6}}}+\cdots ={\frac {\pi ^{6}}{960}}\!$

$\pi ={1}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}-{\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}+{\frac {1}{9}}-{\frac {1}{10}}+{\frac {1}{11}}+{\frac {1}{12}}-{\frac {1}{13}}+\cdots \!$ (Euler, 1748)

Lehenengo bi terminoen ondoan, zeinuak honela zehazten dira: izendatzailea 4m - 1 formako zenbaki lehena bada, zeinua positiboa da; izendatzailea 4m + 1 formako zenbaki lehena bada, negatiboa da; zenbaki konposatuen bidez, zeinua faktoreen zeinuen biderkaduraren berdina da.

Machin-en formulak

${\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}\!$ (Machinen jatorrizko formula)

${\frac {\pi }{4}}=\arctan 1$

${\frac {\pi }{4}}=\arctan {\frac {1}{2}}+\arctan {\frac {1}{3}}\!$ (Eulerrena)

${\frac {\pi }{4}}=2\arctan {\frac {1}{2}}-\arctan {\frac {1}{7}}\!$ (Hermann-ena)

${\frac {\pi }{4}}=2\arctan {\frac {1}{3}}+\arctan {\frac {1}{7}}\!$ (Huttonena edo Vegarena)

${\frac {\pi }{4}}=5\arctan {\frac {1}{7}}+2\arctan {\frac {3}{79}}\!$

${\frac {\pi }{4}}=12\arctan {\frac {1}{49}}+32\arctan {\frac {1}{57}}-5\arctan {\frac {1}{239}}+12\arctan {\frac {1}{110443}}\!$

${\frac {\pi }{4}}=44\arctan {\frac {1}{57}}+7\arctan {\frac {1}{239}}-12\arctan {\frac {1}{682}}+24\arctan {\frac {1}{12943}}\!$

${\frac {\pi }{2}}=\sum _{n=0}^{\infty }\arctan {\frac {1}{F_{2n+1}}}=\arctan {\frac {1}{1}}+\arctan {\frac {1}{2}}+\arctan {\frac {1}{5}}+\arctan {\frac {1}{13}}+\cdots \!$

non $F_{n}$ baita enegarren Fibonacci-ren zenbakia.

Serie infinitu batzuk

Hona hemen pi-rekin lotutako serie infinitu batzuk:

$\pi ={\frac {1}{Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {((2n)!)^{3}(42n+5)}{(n!)^{6}{16}^{3n+1}}}\!$
$\pi ={\frac {4}{Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(4n)!(21460n+1123)}{(n!)^{4}{441}^{2n+1}{2}^{10n+1}}}$
$\pi ={\frac {4}{Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(6n+1)\left({\frac {1}{2}}\right)_{n}^{3}}{{4^{n}}(n!)^{3}}}\!$
$\pi ={\frac {32}{Z}}\!$	$Z=\sum _{n=0}^{\infty }\left({\frac {{\sqrt {5}}-1}{2}}\right)^{8n}{\frac {(42n{\sqrt {5}}+30n+5{\sqrt {5}}-1)\left({\frac {1}{2}}\right)_{n}^{3}}{{64^{n}}(n!)^{3}}}\!$
$\pi ={\frac {27}{4Z}}\!$	$Z=\sum _{n=0}^{\infty }\left({\frac {2}{27}}\right)^{n}{\frac {(15n+2)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{3}}\right)_{n}\left({\frac {2}{3}}\right)_{n}}{(n!)^{3}}}\!$
$\pi ={\frac {15{\sqrt {3}}}{2Z}}\!$	$Z=\sum _{n=0}^{\infty }\left({\frac {4}{125}}\right)^{n}{\frac {(33n+4)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{3}}\right)_{n}\left({\frac {2}{3}}\right)_{n}}{(n!)^{3}}}\!$
$\pi ={\frac {85{\sqrt {85}}}{18{\sqrt {3}}Z}}\!$	$Z=\sum _{n=0}^{\infty }\left({\frac {4}{85}}\right)^{n}{\frac {(133n+8)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{6}}\right)_{n}\left({\frac {5}{6}}\right)_{n}}{(n!)^{3}}}\!$
$\pi ={\frac {5{\sqrt {5}}}{2{\sqrt {3}}Z}}\!$	$Z=\sum _{n=0}^{\infty }\left({\frac {4}{125}}\right)^{n}{\frac {(11n+1)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{6}}\right)_{n}\left({\frac {5}{6}}\right)_{n}}{(n!)^{3}}}\!$
$\pi ={\frac {2{\sqrt {3}}}{Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(8n+1)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{9}^{n}}}\!$
$\pi ={\frac {\sqrt {3}}{9Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(40n+3)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{49}^{2n+1}}}\!$
$\pi ={\frac {2{\sqrt {11}}}{11Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(280n+19)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{99}^{2n+1}}}\!$
$\pi ={\frac {\sqrt {2}}{4Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(10n+1)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{9}^{2n+1}}}\!$
$\pi ={\frac {4{\sqrt {5}}}{5Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(644n+41)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}5^{n}{72}^{2n+1}}}\!$
$\pi ={\frac {4{\sqrt {3}}}{3Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(28n+3)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{3^{n}}{4}^{n+1}}}\!$
$\pi ={\frac {4}{Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(20n+3)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{2}^{2n+1}}}\!$
$\pi ={\frac {72}{Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(4n)!(260n+23)}{(n!)^{4}4^{4n}18^{2n}}}\!$
$\pi ={\frac {3528}{Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(4n)!(21460n+1123)}{(n!)^{4}4^{4n}882^{2n}}}\!$

non

$(x)_{n}\!$

beheranzko faktorialeko Pochhammerren ikurra da.

Produktu infinituak

(Euler)${\frac {\pi }{4}}={\frac {3}{4}}\cdot {\frac {5}{4}}\cdot {\frac {7}{8}}\cdot {\frac {11}{12}}\cdot {\frac {13}{12}}\cdot {\frac {17}{16}}\cdot {\frac {19}{20}}\cdot {\frac {23}{24}}\cdot {\frac {29}{28}}\cdot {\frac {31}{32}}\cdots \!$

non zenbakitzaileak zenbaki lehenak baitira; eta izendatzaile bakoitza zenbakitzailetik hurbilen dagoen 4ren multiploa baita.

$\prod _{n=1}^{\infty }{\frac {4n^{2}}{4n^{2}-1}}={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdots ={\frac {4}{3}}\cdot {\frac {16}{15}}\cdot {\frac {36}{35}}\cdot {\frac {64}{63}}\cdots ={\frac {\pi }{2}}\!$

Vieèteko formula:

${\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdot \cdots ={\frac {2}{\pi }}\!$

Frakzio jarraituak

$\pi ={3+{\cfrac {1^{2}}{6+{\cfrac {3^{2}}{6+{\cfrac {5^{2}}{6+{\cfrac {7^{2}}{6+\ddots \,}}}}}}}}}$

$\pi ={\cfrac {4}{1+{\cfrac {1^{2}}{3+{\cfrac {2^{2}}{5+{\cfrac {3^{2}}{7+{\cfrac {4^{2}}{9+\ddots }}}}}}}}}}$

$\pi ={\cfrac {4}{1+{\cfrac {1^{2}}{2+{\cfrac {3^{2}}{2+{\cfrac {5^{2}}{2+{\cfrac {7^{2}}{2+\ddots }}}}}}}}}}\,$

(ikus, halaber, frakzio jarraitua)

Denetarik

$n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\!$ (Stirling-en hurbilketa)

$e^{i\pi }+1=0\!$ (Eulerren identitatea)

$\sum _{k=1}^{n}\varphi (k)\sim {\frac {3n^{2}}{\pi ^{2}}}\!$ (ikus Eulerren Φ funtzioa)

$\sum _{k=1}^{n}{\frac {\varphi (k)}{k}}\sim {\frac {6n}{\pi ^{2}}}\!$ (ikus Eulerren Φ funtzioa)

$\Gamma \left({1 \over 2}\right)={\sqrt {\pi }}\!$ (ikus Gama funtzioaren trambea)

$\pi ={\frac {\Gamma \left({1/4}\right)^{4/3}\mathrm {agm} (1,{\sqrt {2}})^{2/3}}{2}}\!$ (non agm baita aritmetika ertaina-geometrikoa)

$\lim _{n\rightarrow \infty }{\frac {1}{n^{2}}}\sum _{k=1}^{n}(n\;{\bmod {\;}}k)=1-{\frac {\pi ^{2}}{12}}\!$ (non mod baita modulu-funtzioa, n-ren eta k-ren arteko zatiketaren hondarra ematen duena)

(unitate zirkulu baten azalera ebaluatzeko Riemann-en batukaria)$\pi =\lim _{n\rightarrow \infty }{\frac {4}{n^{2}}}\sum _{k=1}^{n}{\sqrt {n^{2}-k^{2}}}$

$\pi =\lim _{n\rightarrow \infty }{\frac {2^{4n}}{n{2n \choose n}^{2}}}$ (Stirling-en hurbilketaren bidez)