Lista de fórmulas que involucran π

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La siguiente es una lista de fórmulas importantes que involucran la constante matemática π. Muchas de estas fórmulas se pueden encontrar en el artículo Pi o en el artículo Aproximaciones de π.

Geometría eucaclidiana

\pi ={\frac {C}= {\fn} {\fnK}} {\fn}} {\fn}} {\fn}} {\fn}}} {\fn}}} {\f}}} {\fn}}}} {\fn}}} {\f}}} {\f}}}}}}} {\f}}}}}}}}}}}}} {\f}}}}}}}}}}}}}}} {\f}}}}}}}}}}}} {\f}}}}}} {\f}}}} {\f}}}}} {\f}}}}}}}}} {\f}}}} {\f}} {\f}}}} {\f}}} {\f}}}}} {\f}}}} {\f} {\f}}}}}}}}}}}}}}}} {\f}}}} {\f}}}}}}}}}}}}

donde $C$ es la circunferencia de un círculo, $d$ es el diámetro y $r$ es el radio. En términos más generales,

\pi ={\frac {} {}}

donde $L$ y $w$ son, respectivamente, el perímetro y el ancho de cualquier curva de ancho constante.

A=\pi r^{2

donde $A$ es el área de un círculo. En términos más generales,

A=\pi ab

donde $A$ es el área encerrada por una elipse con semieje mayor $a$ y semieje menor $b$ .

{\displaystyle C={\frac {2\pi}{\operatorname {agm}\left(a_{1}{2}-\sum ¿Por qué?

Donde $C$ es la circunferencia de un elipse con eje semi-major $a$ y eje semi-minor $b$ y $A_{n},b_{n$ son las iteraciones aritméticas y geométricas de $\operatorname {agm} (a,b)$ , la media aritmética-geométrica de $a$ y $b$ con los valores iniciales $A_{0}=a$ y $B_{0}=b$ .

A=4\pi r^{2

donde $A$ es el área entre la bruja de Agnesi y su línea asintótica; $r$ es el radio del círculo que la define.

A={\frac {\fnMicrosoft Sans Serif} {\fnMicrosoft Sans {\fnMicrosoft Sans Serif} {\fnMicrosoft Sans Serif} {\fnMicrosoft Sans Serif} {\fnMicrosoft Sans Serif} {\fnMicrosoft Sans} {\fnMicrosoft Sans Serif}} {\fnMicrosoft}}}} {\f}}}}\f}}}}}}}}}}}}}\f}\f}}} {\f}\fnMisq\f}\f}}} {\fnK\f}}}\f}}\f}}}}\f}}}}\fnun\fnun\fnun\fnun\fnun\fnun\fnun\fnun\fnun\fnun\fnun\fnun\fnun\fnun\fnun\fnun\fnun\fnun}}\fnun} }r^{2}={\frac {\pi} {\fnMicrosoft Sans}}

Donde $A$ es el área de un squircle con radio menor $r$ , ${\displaystyle "Gamma"$ es la función gamma.

A=(k+1)(k+2)\pi r^{2

Donde $A$ es el área de una epicicloide con el círculo más pequeño del radio $r$ y el círculo más grande del radio $kr$ () $k\in \mathbb {N$ ), suponiendo que el punto inicial está en el círculo más grande.

A={\frac {}{k}+3}\pi a^{2

Donde $A$ es el área de una rosa con frecuencia angular $k$ () $k\in \mathbb {N$ ) y amplitud $a$ .

L={\frac {\fnMicrosoft Sans Serif} {\fnMicrosoft Sans Serif} ¿Qué? {}

donde $L$ es el perímetro de la lemniscata de Bernoulli con distancia focal $c$ .

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

donde $V$ es el volumen de una esfera y $r$ es el radio.

SA=4\pi r^{2

donde $SA$ es el área de la superficie de una esfera y $r$ es el radio.

H={1 \over 2}\pi ^{2}r^{4

donde $H$ es el hipervolumen de una 3-esfera y $r$ es el radio.

SV=2\pi }r^{3

donde $SV$ es el volumen de la superficie de una esfera tridimensional y $r$ es el radio.

Poligones convexos regulares

Suma $S$ de los ángulos internos de un polígono convexo regular de $n$ lados:

S=(n-2)\pi

Área $A$ de un polígono convexo regular con $n$ lados y longitud de lado $s$ :

A={\frac {\fns}{4}\cot {\fnMic {\pi} } {n}

Radio interno $r$ de un polígono convexo regular de $n$ lados y longitud de lado $s$ :

r={\frac {\fnMicroc} {\fnMicroc} {\fnMicroc} {\fnMicroc}}} {\fn}}}\fn} {\fn}\fn}\fn}\fn}\fn}\fn}\f}\fn\f}\f}\fn\fn\f}\fn}\f}\f}\f}\f}\f}\f}\f}\f}\f}\f}\f}\f}\f}\f}\f}\f}\f}\f}\f}\fn\fn\fn\fn\f}\f}\f}\f}\f}\f}\fn\fn\f}\fn\f}\f}\f}\f}\fn\fn\fn\f}\f}\fn\fn\fn\fn\fn\fn\fn\fn\fn\f}\f}\f } {n}

Radio circunscrito $R$ de un polígono convexo regular de $n$ lados y longitud de lado $s$ :

R={\frac {}{2}\csc {\fnMicroc} } {n}

Física

La constante cosmológica:

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

Principio de incertidumbre de Heisenberg:

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

Ecuación de campo de Einstein de la relatividad general:

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

Ley de Coulomb para la fuerza eléctrica en vacío:

{\displaystyle F={\frac {\4\fnK} {\cHFF} {\4\cH00}\fnK} {\cHFF} {\cH00}} {\4\fn}}\fn}}}}\fn9\fn\fn\fn} \varepsilon ¿Qué?

Permeabilidad magnética del espacio libre:

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

Período aproximado de un péndulo simple con pequeña amplitud:

T\approx 2\pi {\fnMicroc} {} {}}}

Exacto período de un péndulo simple con amplitud ${\displaystyle \theta ¿Qué?$ () $\operatorname {agm$ es la media aritmética-geométrica:

T={\frac {2\pi}{\operatorname {agm} (1,\cos(\theta _{0}/2)}}}{\sqrt {\frac {} {}}}

La tercera ley de Kepler del movimiento planetario:

{\frac {\f} {\fnK}}={\f}}={\f}} {\f}} {\f} {\f}}}} {\f}}}} {\f}}}}} {\f} {GM}{4\pi ^{2}

La fórmula de pandeo:

{\displaystyle F={\frac {\fnMicroc} .

Un rompecabezas que involucra "bolas de billar que chocan":

\lfloor {b^{N}\pi}\rfloor

es el número de colisiones que se producen (en condiciones ideales, perfectamente elásticas y sin fricción) entre un objeto de masa m inicialmente en reposo entre una pared fija y otro objeto de masa b^2Nm, al ser golpeado por el otro objeto. (Esto da los dígitos de π en base b hasta N dígitos después del punto de la base.)

Fórmula que produce π

Integrales

2\int _{-1}{1}{\sqrt {1-x^{2}\,dx=\pi

(integrando dos mitades

y(x)={\sqrt {1-x^{2}}

para obtener el área del círculo de unidad)

\int _{-\infty }\infty }\operatorname {sech} x\,dx=\pi

\int _{-\infty }{\infty }\int _{t}^{\infty }e^{-1/2t^{2}-x^{2}\,dx\,dt=\int ¿Qué? }e^{-t^{2}-1/2x^{2}+xt}\,dx\,dt=\pi

\int ¿Por qué? {1-x^{2}}=

\int _{-\infty . {dx}{1+x^{2}=\pi}

(ver también la distribución Cauchy)

\int _{-\infty }{\infty }{\frac {\sin x}{x}}\,dx=\pi

(véase Dirichlet integral)

\int _{-\infty }e^{-x^{2}\,dx={\sqrt {\pi}

(ver Gaussian integral).

{\displaystyle \oint {\frac {dz}{z}=2\pi} i

(cuando el camino de la integración vientos una vez contrarreste alrededor de 0. Vea también la fórmula integral de Cauchy).

\int _{0}{\infty }\ln \left(1+{\frac {1}{x^{2}}}\right)\,dx=\pi

\int _{-\infty }{\infty }{\frac {\sin x}{x}}\,dx=\pi

{\displaystyle \int ¿Por qué?

(ver también Prueba de que 22/7 excede π).

{\displaystyle \int ¿Por qué?

{\displaystyle \int _{0}{\infty}{\frac {\cHFF} {\cH00FF} -1}{x+1}\,dx={\frac {\pi}{\sin \pi \alpha }, \quad 0 observado \alpha .

{\displaystyle \int _{0}{\infty}{\frac {dx}{\sqrt {x(x+b)}}}={\frac {\pi }{\operatorname {agm} ({\sqrt {a}} {\sqrt {b}}}}}}}}}}}}}}}}}}}}}} {\

(donde)

\operatorname {agm

es el medio aritmético-geométrico; vea también elíptico integral)

Note que con componentes simétricos $f(-x)=f(x)$ , fórmulas de la forma ${\textstyle \int _{-a}{a}f(x)\,dx}$ también se puede traducir a fórmulas ${\textstyle 2\int _{0}{a}f(x)\,dx}$ .

Serie infinita eficiente

\sum _{k=0} {\infty}{\frac {k}{(2k+1)!}=\sum! ¡No! {\cHFF} } {2}

(ver también Doble factorial)

\sum _{k=0}{\infty }{\frac {k}{2^{k}(2k+1)!}={\frac {2\pi }{3{\sqrt {}}

\sum _{k=0}{\infty }{\frac {k!\,(2k)!\,(25k-3)}{(3k)! {\fnMicrosoft Sans Serif}= {\fnMicroc} } {2}

\sum _{k=0}{\infty }{\frac {(-1)^{k}(6k)!(13591409+545140134k)}{(3k)}^{3}640320^{3k}}}={\frac {4270934400}{\sqrt {10005}}}}}} {\

(ver algoritmo de Chudnovsky)

{\displaystyle \sum _{k=0}{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4} {4k}}=\frac {9801}{2{\sqrt {2}}\pi }}}}}}}} {\f}}} {\f}}}}} {\

(véase la serie Srinivasa Ramanujan, Ramanujan–Sato)

Los siguientes son eficientes para calcular dígitos binarios arbitrarios de $π$ :

\sum _{k=0}{\infty }{\frac {(-1)^{k}{4^{k}}}}\left({\frac} {2}{4k+1}+{\frac} {2}{4k+2}+{\frac {1}{4k+3}\right)=\pi)

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

(ver fórmula Bailey–Borwein–Plouffe)

\sum _{\infty }{\frac {1}{16^{k}}}\left({\frac} {8}{8k+2}+{\frac {4}{8k+3}+{\frac} {4}{8k+4}-{\frac} {1}{8k+7}\right)=2\pi

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

Serie de Plouffe para calcular dígitos decimales arbitrarios de $π$ :

\sum _{k=1}{\infty }k{\frac {2^{k}k}{2 k)}{(2k)}}=\pi +3

Otras series infinitas

\zeta (2)={\frac {1}{2}}+{\frac} {2}{2}}+{\frac} {1}{2}}+{\frac} {1}{4^{2}}+\cdots ={\frac {\pi ^{2}{6}}}

(véase también el problema de Basilea y la función Riemann zeta)

\zeta (4)={\frac {1}{4}}+{\frac} {2}{4}}+{\frac} {1}{4}}}+{\frac {1}{4}}+\cdots ={\frac {\pic} ^{4} {90}

\zeta (2n)=\sum _{k=1}{\infty }{\frac {1}{2n}\,={\frac} {1}{2n}}+{\frac} {2}{2n}}+{\frac} {1}{2n}}+{\frac} {1}{2n}}+\cdots =(-1)^{n+1}{\frac {B_{2n}(2\pi)}{2n}{2n}}}}}}}}} {2n}}}} {2n}}}}}}} {2n}}}}}} {2n}}}}}} {2n}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {2n}}}{2n}{2n}}}} {2n}}}}}}}}}}{2n}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}

Donde B_2n es un número de Bernoulli.

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

\sum _{n=1}{\infty }{\frac {7^{n}{8^{n}}}\,\zeta (n+1)=(1+{\sqrt {2})\pi

\sum _{n=2}{\infty }{\frac {2(3/2)^{n}{n} {\zeta (n)-1)=\ln\pi

{\displaystyle \sum _{n=1}{\infty }\zeta (2n){\frac {x^{2n} {n}=\ln} {\fnMic {\pi x}{\sin\pi x}}\quad 0 Segnunció a la muerte

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

(ver fórmula Leibniz para pi)

\sum _{n=0}{\infty }{\frac {(-1)^{(n^{2}-n)/2}{2n+1}=1+{\frac {1}{3}-{\frac {1}{5}-{\frac} {1}{7}+{\frac} {1}{9}+{11}-\cdots ={\frac {\pic} }{2{\sqrt {2}}}

(Newton, Segunda carta a Oldenburg, 1676)

{3}\fn} {3}\fn} {\fn} {\fn} {\fn} {\fn} {\fn} {\fn} {\c}\c} {\c}\c}\c} {\c\c}\c} {\c}\c\c} {\c}\c} {3\c}\c} {3\c}\c}\c}\c}\c}\c\c\c}\c\c}\c\c\c}\c\c\c}\c\c} {\c\c\c}\c\c\c\c\c\c\c\c\c}\c\c\c\c}\c\c\c\c\c\c\c\c\c}\c\c\c\c\c\c\c\c\c}\c\c\c\c\c\c\c\c\c\c\c\c\c}\c\c}\c}\c}\c {1}{\sqrt {3}={\frac {\pi}{2{\sqrt {}}}

(Medhava series)

\sum _{n=1}{\infty }{\frac {(-1)^{n+1}{n^{2}}}={\frac} {\frac} {1}{2}}-{\frac} {2}{2}}+{\frac} {1}{3^{2}}-{\frac {1}{4}}+\cdots ={\frac {\pic} ^{2}{12}

{\displaystyle \sum _{n=1}{\infty }{\frac {1}{2n)}}}={\frac {1}{2}}}}+{\frac {1}{4}{2}}}}}}+{\frac {2}}{6}}{2}}}}}} {\frac}}}{\frac}{\frac}}{\frac}{\f}}}}}{\f}{\f}}{\f}}}}{\f}{\f}}{\f}{\f}}}}}}}{\f}}}}}{\f}}}{\f}}}}{\f}}}}}{\f}}}}}}}}}}{\f}}}}{\f}}{\f}}}}}}}}}}}}{\f}{\f}}}}{\f}}}}}}}}}}}}}}} {1}{8^{2}}+\cdots ={\frac {\pi .

\sum _{n=0} {\infty }\left({\frac {1}{2n+1}}\right)^{2}={\frac {1}{2}}+{\frac} {1}{2}}+{\frac} {1}{5^{2}}+{\frac {1}{7^{2}}+\cdots ={\frac {\pic} ^{2} {8}

\sum _{n=0}{\infty }\left({\frac {(-1)^{n}{2n+1}\right)}{3}={\frac {1}{1} {3}} {\frac {1}{3}}}+{\frac {1}{5^{3}}}}-{\frac {1}{7^{3}}}+\cdots ={\frac {\pic {\pic {\pic}}} {\cdots} {\cdots}}} {\cdots} {\cdots} {\cdots}}}} {\cdots} {\c}} {\c} {\cdots} {\cdots} {\c} {\c}}} {\c}}}}} {\c}}} {\f} {\f} {\cdots} {\cdots} {\cdots} {\fn\c}}} {\c}}}}}}}} {\c}}}}}} {\cdot}}} {\cdots} {\cdot}}} ^{3} {32}

{\displaystyle \sum _{n=0}\infty }\left({\frac {1}{2n+1}}\right)^{4}={\frac {1}{4}}+{\frac {1}{4}}+{\frac {1}{5^{4}}}+{\frac {1}{7} {4}}}}+\cdots ={\frac {\pic {\pic}} {\cdots}} {\cdots} {\cdots}} {\cdots={\cdots} {\f}} {\cdots}}}}}} {\c} {\cdots}}} {\c} {\cdots {\c} {\c}}} {\c}}}}}}{\f}}}}{\f} {\f} {\cdots} {\f} {\cdots}} {\f}} {\f}}} {\f}}}}}}}}}} {\cdot}}}}\cdots}}\cdots} {\cdot}} .

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

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

En general,

{\fnMicrosoft Sans Serif} {\fn} {\fn} {\fn}{2n}{2k+1}}}=(-1)}{k}{k}{2k}{2k}{2k}{2k}}}}}\left({\bi }{2}{2}}}\right)} {\c} {\c}\c}}\c}}}}}}\c}\c}}\c}\c}\c}\c}\c}\c}\c}\c}\c}\c}\c}\c\c\c\c\c\c\c\c\c\c}\c\c}\c\c\c\c\c\c\c\c\c}\c\c\c\c\c\c\c\c}\c\c\c\c\c\c\c\c\c\c\c\c\c\c\c\c\c\c}\c\c\c {N}

Donde $E_{2k$ es $2k$ El número de Euler.

\sum _{\n=0}{\infty}{\binom {\frac {1}{n}{n}{\frac {1}{n}{2n+1}=1-{\frac {\fn} {\fn} {\fn}} {\fn}} {\fn} {\fn} {\fn} {\fn} {\fn} {\fn} {\fn}}} {\fn}} {\fn} {\fn}}} {\fn}\fn}}}}\fn} {\fn} {\fn} {\fn}} {\fn}\fn} {\fn} {\fn}\fn}\fn}\fn}\fn}\fn} {\fn}\fn} {\fn} {\fn}\fn} {\fn}\fn}\fn}\fn}\fn}\fn} {\fn}\fn}\fn}\fn}\fn}\fn}\fn {1}{6}-{\frac} {1}{40}-\cdots ={\frac {\pi } {4}

\sum _{n=0}{\infty }{\frac {1}{(4n+1)(4n+3)}}}={\frac {1}{1\cdot 3}+{\frac {1}{5\cdot 7}+{9\cdot 11}{9\cdot 11}}\cdot {\cdot\cdot\cdot} {\cdot} {\cdot\cdot}{\cdot\cdot} {\cdot}{\cdot\cdot}{\cdot}{\cdot\cdot\cdot}{\cdot\cdot} {\cdot} {} {\cdot} {\cdot {\c}{\cdot}{\cdot\c}{\cdot\c}{\cdot\cdot\cdot}{\cdot}{\c} {\cdot} } {8}

{\fnMicrosoft Sans Serif} {\fn0}} {\fn1}\n0}\n1}\n0}\n0}\n0}\n0}\n1}\n1}\n1}\n1}\right=Principalmentenncipalmente, no es posible. ={\frac {\sqrt {3}{\pi} }

(ver coeficientes Gregory)

\sum _{\n=0}{\infty}{\frac {(1/2)_{n} {2} {2}n} {2}}}\sum}} {2}}}}} {2}}} {2}}} {2} {\n}{n}}}}}}}}}}}}}}}}}}}}}} {2}{2} {\n}} {\n} {\n} {\n}}}} {\n}}}}}}}}}}}}}}}}{2}}}}}}}}}}}{2} {\n} {\n}} {\n}}}} {\n}}}}}}}}}}}}}}}}}}}}}{2}}}}{2}}}}}}}}}}}}}}}}}}{2}}}}}}{2}}}{2}}}}}}}}}}}}}}}}}}}}}}}}}}}}}{2}}}}}}}} {\ ¿Por qué? {1}{\pi}}

(donde)

(x)_{n

es el factorial creciente)

\sum _{n=1}{\infty }{\frac {(-1)^{n+1}{n(n+1)(2n+1)}}=\pi -3

(Nilakantha series)

\sum _{n=1}{\infty }{\frac {\fn} {\fn} {\fn}} {\binom}} {\fn}} {\fn} {\fn}} {\fn}}}} {\fn}}} {\fn}}}} {\fn}}}} {\fn}}} {\fn}}}} {\binom}}}}} {\f}}}}}}}}}}}}}} {\binom}} {\ {2n} {\fn}={4\pi}{2}{25{\sqrt {}}

(donde)

F_{n

es n- el número Fibonacci)

\sum _{n=1} {\infty }\sigma (n)e^{-2\pi n}={\frac {1}{24}-{\frac {1}{8\pi}

(donde)

\sigma

es la función suma de visores)

\pi =\sum _{n=1}{\infty }{\frac {(-1)^{\epsilon (n)}{n}}}=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

(donde)

\epsilon (n)

es el número de factores principales de la forma

p\equiv 1\,(\mathrm {mod} \,4)

n

)

{\frac {\fn\fnMicrosoft {\fnMicrosoft {\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\fnMicrosoft {\\\\\\fn\\\\\\\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\\\\\\\\\\\\fn\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\fnMin ##{2}=\sum _{n=1}{\infty }{\frac {(-1)^{\varepsilon (n)}{n}}}=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}+\cdots

(donde)

\varepsilon (n)

es el número de factores principales de la forma

p\equiv 3\,4

n

)

\pi =\sum _{n=-\infty }{\infty }{\frac {(-1)^{n}{n+1/2}} {} {\fn}} {\fn0}} {\fn0}}

\pi ^{2}=\sum _{n=-\infty }{\infty }{\frac {1}{(n+1/2)^{2}}} {} {\fn0} {\fn0}} {\fn0}

Las dos últimas fórmulas son casos especiales de

{\fn} {\fn} {\fn}\fn}} {\fn}} {\fn} {\fn} {\fn} {\fn} {\fn} {\fn} {\fn} {\fn}\fn} {\fn}\fn}\fn\fn\fn\fn} {\fn\fn}}}}\fn}}}\fn}\fn}\fn\fn\fn}\fn\fn}\fn\fn}\fn}\c\fn\fn}\c\fn\fnK\c\fn}\fn\fn}\fn}\fn}\fn}\c\c\c\c\c\c\c\c\cH00}\c\c\c\c\c\fn}\cH00}}\cH00}}\c\cH00}\c\c\c\c\c\c\c

que generan infinitamente muchas fórmulas análogas para $\pi$ cuando $x\in \mathbb {Q} \setminus \mathbb {Z$

Aquí se dan algunas fórmulas que relacionan a $π$ con los números armónicos. Otras series infinitas que involucran a π son:

$\pi ={\frac {1}{Z}$	$Z=\sum _{n=0}{\infty }{\frac {(2n)!)^{3}(42n+5)}{(n)}{6}{6}{3n+1}}}}}}} {}}}} {\fn0}} {\fn0}} {\fn0}}} {\f}}}}}} {\f}}}}}}}}}}}}} {\f}}}}}} {\f}}}}}}}}}} {\f}}}}}} {\f}}}}} {\f}}}}}}}}}} {\f}}}}}}}}}}} {\f}} {\f}}}}}}}}}}}} {\f}}}}}}}}} {\f}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$
$\pi ={\frac {4} {Z}}$	${\displaystyle Z=\sum _{n=0}{\infty }{\frac {(-1)^{n}(4n)!(21460n+1123)}{(n)}{4}{4}{2n+1}{2}{2}{10n+1}}}}}}}}}}}} {\$
$\pi ={\frac {4} {Z}}$	$Z=\sum _{n=0}{\infty }{\frac {(6n+1)\left({\frac {1}{2}}\right)_{n}{4}{n} {\n} {\n} {\n}}}} {}}}}}}}} {\n} {\n}}} {\n}}}} {\n}}}}}}} {\n} {\n}}}}}}}}}}} {\n} {\n}}}}} {\n}}}}}}}}}}}}}}} {\n} {\n} {\n} {\n}}}}} {\n} {\n}} {\n}}}}}}}}}}} {\n}}}}}}}} {\n}}}}}}}}}}}}}}}}}}}}}}} {\n}}}}}}}}}}}}}$
$\pi ={\frac {32} {Z}}$	$Z=\sum _{\n=0}{\infty }\left({\frac {\sqrt {5}-1}{2}\right)}{8n}{\frac {42n{\sqrt {5}}+30n+5{\sqrt {5}-1)\left({\frac {1}{2}}\right)_{n}}{3}{64^{n} {\n} {3}}}}}} {}}}}}} {}}}}}} {}}}}}}}} {}}}} {}}}}}}} {}}}}}}}}} {}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$
${\fnK}$	${\displaystyle Z=\sum _{n=0}{\infty }\left({\frac {2}{27}\right)}{n}{n}{\frac {15n+2)\left({\frac {1}{2}\right)_{n}\left({\frac {\frac}{3}\right)}{\n} {\n}{\n}\n}\n}}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n\n}\n}\ {2} {3}} {n} {n} {n}}}} {\cH00}} {\cH00}}} {\cH00}} {\cH00}}}}} {\cH00}}}}} {\cH00}}}} {\cH00}}}}}} {\cH00}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {\c\c\c\c}}}}}}}}}}}}}}}}}}}}}} {\c\c}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {\c\c}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {\$
$\pi ={\frac {15{\cH00} {3}} {2Z}}} {}}} {}}}} {\c}}}}} {\c}}}}}}}}}}}}}} {}}}}}}} {}}} {}}}}}}}}} {}}}}}}}}} {}}}}}}}}}}}}} {}}}}}}}} {}}}}}}}}}}}}}}}}} {}}}}} {}}}}}}}}}} {}}}}}}} {}}}}} {}}}}}}}}} {}}}}}}}}} {}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {}}}}}} {}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}}$	${\displaystyle Z=\sum _{n=0}{\infty }\left({\frac {4}{125}}\right)}{n}{n}{\frac {(33n+4)\left({\frac {1}{n}\right)_{n}\left({\frac {\frac}{3}\right)}{\n}{\n}{\n}{\n}\n}\n}{\n}\n}\n}\n}\n}{\n}\n}\n}{\n}\n}\splaystyle Z=0}\splaystyle Z=0}\splaystyle Z=\s0} {\s0} {\s0} {\show}\s0}{\n}}\fn}\s0}{\fn}}\fn}}\fn} {\fn}\fn}\fn}\f}\fn}\f}\f}\fn {2} {3}} {n} {n} {n}}}} {\cH00}} {\cH00}}} {\cH00}} {\cH00}}}}} {\cH00}}}}} {\cH00}}}} {\cH00}}}}}} {\cH00}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {\c\c\c\c}}}}}}}}}}}}}}}}}}}}}} {\c\c}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {\c\c}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {\$
${85}{18{\sqrt {85}}{18{\sqrt {3}Z}}$	${\fnMicrosoft Sans Serif} {\fn}\fn} {\fn\fnK}}\n}\n}\n} {\fn} {\fn\fn+8)\left(\frac {1}{2}\right)_{n}\m} {\fn} {\fn} {\c} {\c}\c}\c}\c}\c}\c}\c}\c}\c}\c}\c}\c}\c}\c}\c}\c}\c}\c}\c}\c\c\c}\c\c\c\c\c\c\c\c}\c}\c}\c}\c\c}\c\c\c}\c\c\c\c}\c\c\c\c\c}\c}\c}\c}\c}\c\c\c\c}\c\c\c}\c\c\c}\c}\c\ {5} {6}} {n} {n}{n}}} {\cH00}}} {\cH00}}} {\cH00}}} {\cH00}}}} {\cH00}}}}}}} {\cH00}}}} {\cH00}}}}}} {\cH00}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {\c}}}}}}}}}}}}} {\c}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {\$
$\pi ={\frac {5{\sqrt {5}}{2{\sqrt {3}Z}}$	${\fnK} {\fn\fn\fn\fn\fn\fn\fn\fnn\fn}\nn} {\fn\fn+1)\left({\frac {2}{n}\n}\n}\n}\n}\m}\c\c\cHFF} {\c\c}\cH00}\c}\cH00}\fn}\fn}\c\c}\c\c\c}\c\c}\c}\c}\c}\c\c\c\c\c\c\c\c\c\c\c\cH00}\c\c\c}\c\c\c\c\c\c\c\c\c\c\c}\c\c\c\c\c\c\c}\c}\c\cH00}\cH00}\c\cH00}\c\cH0\c}\c\c}\c}\c}\c {5} {6}} {n} {n}{n}}} {\cH00}}} {\cH00}}} {\cH00}}} {\cH00}}}} {\cH00}}}}}}} {\cH00}}}} {\cH00}}}}}} {\cH00}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {\c}}}}}}}}}}}}} {\c}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {\$
$\pi ={\frac {2{\sqrt {}} {Z}}} {}}} {}}} {}}}}} {}}}} {}}}} {}}} {}}}}}} {}}}} {}}}} {}}}}} {}}}} {}}}}} {}}}}}} {}}}}}}}}}}} {}}}}}}}} {}} {}}}}}} {}}}}}}}}}}}} {}}}}}}}}}}} {}}}}}}}}}}}}}}}} {}}}}} {}}}}}}}}}}}}} {}}}}}}}}}} {}}}}} {}}}}}} {}}}}}}} {}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$	$Z=\sum _{n=0}{\infty }{\frac {(8n+1)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}\right)_{n}\left({\frac}{\frac}{\frac} {\f}\f}\f}\f}}\f}\fn}\f}\fn}\f}\f}\f}\f}\f}\fn}\fn}\c}\fn}\fn}\c}\fn}\fn}\fn}\c}\fn}\fn}\fn}\c}\p] {3}} {}} {n} {n} {} {}}} {}}}} {}}} {}}}} {}}} {}}}} {}}} {}} {}}}}}}} {} {}}}}}}}}}}} {}} {}}}} {}}}} {}}}}}} {}}}}}}}}}}}}}}}}}}}}}}}} {}}}} {}}}}}} {}}}}} {}}}}}}}}} {}}}}} {}}}}}} {}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}}}}}}}}}$
$\pi ={\frac {\cHFF} {3}{9Z}} {}} {}}} {}}} {}}}}} {}}}}} {}}}}} {}} {}}}} {}}}}} {}}}} {}}}} {}}}}}} {}}}}} {}}}}} {}}}}} {}}}}}} {}}}}}} {}}}}}}}}}}}}}} {}}}}}}}}}}}} {}}}}}}}}}}}} {}}}}}}}} {}}}}}}}}}}}} {}}}}}}}}}}}}}}} {}}}}} {}}}}}}}} {}}}}}}} {}}}}}}}}}}} {}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}}}}}}}}$	$Z=\sum _{n=0}{\infty }{\frac {(40n+3)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}\right)_{n}\left({\frac}\frac}{\frac}{\n}\n}\n}\n}\n}{\n}{\n}\n}\n}\n}\n}\n}{\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n} {3}{4}} {n} {n} {3}{3}{2n+1}}}}}}$
$\pi ={\frac {2{\sqrt {11}{11Z}}} {\f}} {\f}} {\f}}}} {\f}}} {\f}}}} {\f}}} {\f}}}$	$Z=\sum _{n=0}{\infty }{\frac {(280n+19)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}\right)_{n}\left({\frac\frac} {\n}{\n}{\n}\n}\n}\n}\n}\n}}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\n}\ {3} {4}} {n} {n}{n} {3}{2n+1}}}$
$\pi ={\frac {\cHFF} {2}{4Z}}} {}}} {}}}} {}}}}} {}}}}} {}}}} {}} {}}}}}} {}}}}}}}}} {}}}}}}} {}}}} {}}} {}}}}} {}}}}} {}}}}}}}}}}}} {}}}} {}}}}} {}}}}}}}}}}}}}}}}}}}}}}}}} {}}}}}}} {}}}}}}} {}}}}} {}}}}}}}}}}}}}} {}}}}}}}}} {}}} {}}}}}}}}}}}}}}}} {}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}$	${\displaystyle Z=\sum _{n=0}{\infty }{\frac {(10n+1)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}\right)_{n}\left({\frac}{\frac}{\frac}{\f}\f}\f}\f}\f}\fn}\f}\fn}\f}\f}\f}\f}\f}\f}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\c}\fn}\fn}\fn}\c}\fn}\fn}\c}\c}\c}\c}\c}\fn}\c}\ {3} {4}} {n} {n}{n}} {3}{2n+1}}}}} {\cH00}}}}} {\cH00}}}}}}}} {\cH00}}}}} {\cH00}}}}}} {\cH00}}}}}}}}}} {\c\cH00}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {\c}}}}}}}}}}}}}}}}}}}} {\$
$\pi ={\frac {4{\sqrt {}} {5Z}}} {}}}} {}}}}}}} {}}}}}} {}}}} {}}}}}}}} {}}}}}} {}}}} {}}} {}}}}} {}}}}}}}}}}} {}}}} {}}}}}}}} {}}}} {}}}}}}}}}}}}}} {}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}} {} {} {}}} {}}}}}} {}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}}}}}}}}}$	$Z=\sum _{n=0}{\infty }{\frac {(644n+41)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}\right)_{n}\left({\frac\frac}{\frac} {\f}\f}\f}\f}\f}\fn}\fn}\fn}\fn}\fn}\fn}\f}\fn}\fn}\c}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\c}\fn}\c}\fn}\c}\fn}\fn}\fn}\fn}\c}\c\fn}\c}\fn}\c}\c}\ {3}{4}} {n}{n} {3} {3}{n}{2n+1}}}}}}}$
$\pi ={\frac {4{\sqrt {3}} {3Z}}} {3Z}}} {}}} {3Z}}}} {}}} {}}} {}}}}} {}}}} {}}}}} {}}}} {}}}}}} {}}}} {}}}}}} {}}}}}}} {}}}}}} {}}}}}}} {}}}}}}}} {}}}}}}} {}}}}} {}}}}}}}}}}} {}}}}}}} {}}}}}}} {}}}}}} {}}}}}}}}}}} {}} {}}}}}}} {} {}}}} {}}}}}}}}}}}}} {} {}}}}}}}}}}} {}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}}}}}}}}$	${\displaystyle Z=\sum _{n=0}{\infty}{\frac {(-1)^{n}(28n+3)\left({\frac {1}{2}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac}\frac} {\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\p] {3}{4}} {n} {n}} {3}{4} {4} {n+1}}}}}}} {\n}}}} {\cH00}}}}}} {\cH00}}}}}}}} {\cH00}}}}}}} {\c}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {\$
$\pi ={\frac {4} {Z}}$	$Z=\sum _{n=0}{\infty }{\frac {(-1)^{n}(20n+3)\left({\frac {1}{2}\right)_{n}\left({\frac {1}{4}\right)_{n}\left({\frac}\frac} {\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\fn}\p] {3}{2} {2n+1}}$
$\pi ={\frac {72}{Z}$	$Z=\sum _{n=0}{\infty }{\frac {(-1)^{n}(4n)!(260n+23)}{(n)}{4}4n}18^{2n}}}}}} {}}} {\fn}} {\fn0}}}} {\fn0$
${\fnK}$	${\displaystyle Z=\sum _{n=0}{\infty }{\frac {(-1)^{n}(4n)!(21460n+1123)}{4}4}{4n}882}}}}}}}}}}}}} {\$

Donde $(x)_{n$ es el símbolo Pochhammer para el factorial creciente. Vea también la serie Ramanujan–Sato.

Fórmula tipo mecanizado

{\frac {\fn\fnMicrosoft {\fnMicrosoft {\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\fnMicrosoft {\\\\\\fn\\\\\\\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\\\\\\\\\\\\fn\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\fnMin } {4}=\arctan 1

{\frac {\fn\fnMicrosoft {\fnMicrosoft {\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\fnMicrosoft {\\\\\\fn\\\\\\\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\\\\\\\\\\\\fn\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\fnMin } {4}=\arctan {\fnMicroc} {\fnh}\\fnh}\\fnh\fnh\fnh}\\\fn}\\\cH30}\\\\\\fn\fn}\\\\\fn}\\\\\\fnh}\\\\\\\fnK\fnh}\\\\\\fn\\\\fnhnhnh\\\\\\\\fnhnh\\fn\\\fnK\fnh}\\\\\fnK\fn\\fn}\\\\fn\fnK}\\\\\\\\\\\\\\\\\fnK\fnK\fnh\\\\fnh\\\\\fn\\fnh\fn}\fnh}\\\\fn}\\fnh}\\\\\\\\\\\fn}\\\\\fn {1}{3}

{\frac {\fn\fnMicrosoft {\fnMicrosoft {\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\fnMicrosoft {\\\\\\fn\\\\\\\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\\\\\\\\\\\\fn\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\fnMin ## {4}=2\arctan {\frac {1}{2}-\arctan {\frac} {1}{7}

{\frac {\fn\fnMicrosoft {\fnMicrosoft {\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\fnMicrosoft {\\\\\\fn\\\\\\\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\\\\\\\\\\\\fn\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\fnMin ## {4}=2\arctan {\frac {1}{3}+\arctan {\frac} {1}{7}

{\frac {\fn\fnMicrosoft {\fnMicrosoft {\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\fnMicrosoft {\\\\\\fn\\\\\\\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\\\\\\\\\\\\fn\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\fnMin {4}=4\rctan {\fnMicroc {1}{239

(la fórmula original de Machin)

{\frac {\fn\fnMicrosoft {\fnMicrosoft {\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\fnMicrosoft {\\\\\\fn\\\\\\\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\\\\\\\\\\\\fn\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\fnMin {4}=5\arctan {\fnMicroc} {1}{7}+2\arctan {\frac} {3}{79}}

{\frac {\fn\fnMicrosoft {\fnMicrosoft {\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\fnMicrosoft {\\\\\\fn\\\\\\\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\\\\\\\\\\\\fn\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\fnMin ### {4}=6\arctan {\frac {1}{8}}+2\arctan {\frac {1}{57}+\arctan {\frac {1}{239}

{\frac {\fn\fnMicrosoft {\fnMicrosoft {\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\fnMicrosoft {\\\\\\fn\\\\\\\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\\\\\\\\\\\\fn\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\fnMin ## {4}=12\arctan {\frac {1}{49}+32\arctan {\frac} {1}{57}-5\arctan {\frac} {1}{239}}+12\arctan {\frac {1}{110443}

{\frac {\fn\fnMicrosoft {\fnMicrosoft {\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\fnMicrosoft {\\\\\\fn\\\\\\\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\\\\\\\\\\\\fn\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\fnMin {4}=44\arctan {\fnMicroc} {1}{57}+7\arctan {\frac} {1}{239}-12\arctan {\frac {1}{682}}+24\arctan {\frac {1}{12943}}}}} {}}}}}}}} {}} {}}}} {}}}} {}}}} {}}}}} {}}} {1}}}} {1}}}}} {}}}}}}}}}}}} {}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}} {}}}}}}}} {}}}}}}}}}} {}}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}

Productos infinitos

{\frac {\fn\fnMicrosoft {\fnMicrosoft {\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\fnMicrosoft {\\\\\\fn\\\\\\\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\\\\\\\\\\\\fn\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\fnMin {4}}=\left(\prod _{p\equiv 1{\pmod {4}{\frac {p} {p-1}\right)\cdot \left(\prod _{p\equiv 3{\pmod {4}}}{\frac {p}}{p+1}}}}\right)={\frac {3}{4}\cdot {frac} {5}{4}\cdot {7}\cdot {\cdot {11}{12}\cdot {13}{12}}\cdot {13}\cdot}\cdot}\cdots}\cdot {\cdot}\cdot {\cdot}\cdots}\cdots}\cdots}\cdots}\cdot {\cdots}\cdots}\cdots}\cdots}\cdots} {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdots}\cdot {\cdot {\cdot {\cdot {}\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {} {} {}}

(Euler)

donde los numeradores son los primeros impares; cada denominador es el múltiple de cuatro más cercano al numerador.

{\frac {\fnMicrosoft}\pi} }{6}=\left(\displaystyle \prod _{p\equiv 1{\pmod {6}\atop p\in \mathbb {P} }{\frac {p}\right)\cdot \left(\displaystyle \prod _{p\equiv 5{\pmod {6}\atop p\in \mathbb {P}{\frac} {p}{p+1}\right)={\frac {5}\cdot {\frac} {7}{6}\cdot {11}\cdot {13}{12}\cdot {\cdot {\cdot {17}{18}\cdots }\cdot {\cdot {\cdot} {\cdot {\cdots} {\cdots} {\cdots} {\cdot}} {\cdots}} {\cdot}\cdots}\cdots}\cdots} {\cdots}\cdots} {\cdots}\cdots}\cdot {\cdots}\cdots}\cdot {\cdot {\cdot {\cdots}\cdots}\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot}\cdot {\cdot {\cdot}

{\frac {\fn\fnMicrosoft {\fnMicrosoft {\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\fnMicrosoft {\\\\\\fn\\\\\\\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\\\\\\\\\\\\fn\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\fnMin } {2}=\prod {\c} {\c} {\c} {\c}} {\c}} {\c}} {\c} {}} {\c}} {\c} {\c} {\c}}} {} {\c}} {\c}} {\c}} {}} {\c} {\c}}} {\c}} {\c} {}}} {\c} {} {\c}\c} {\c}\c} {\c}\c}}}}}} {}}} {} {\c}\c} {\c}\c} {\c} {\c} {\c}\c}\c}\c}}} {}}}}}} {\c} {\c} {\c} {\c}} {\c}\c} {\c} {\c}\c}\c} {\c}\c} {\c}\c}\c}\c}\c}\c}\c}\c}\c}}}}}

(ver también producto Wallis)

{\frac {\fn\fnMicrosoft {\fnMicrosoft {\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\fnMicrosoft {\\\\\\fn\\\\\\\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\\\\\\\\\\\\fn\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\fnMin } {2}=\prod ¿Por qué? {1}{1}\derecha)}{+1}\left(1+{\frac {1}{2}\derecha)}{-1}\left(1+{\frac {1}{3}\derecha)}{+1}\cdots

(Otra forma de producto de Wallis)

La fórmula de Viète:

{\frac {2}{\f} {\fnMicrosoft}} {\fnMicrosoft}} {\f}}} {\f}}}}\f}\fnK}}} {\fnK}}}}}}}} {\f}\fn\f}}\fn\fnK\f}}} {\fnMicroc {\fnK}}\cdot {\fnMicroc {\sqrt} {\c}} {\cdot {\fn}\cdot {\fn\sqrt {\sqrt {\cHFF} {\c}}} {\cdot}}} {\cdot} {\cdot {\c}}}}\cdot {\cdot {\cdot {\cdot {\cdot {\f}\f}\f}\cdot {\f}\cdot {\f}\fn\fn\f}\f}\f}\f}\f}\f}\fn\fn\c\fn\fn\fn\fn\fn\fn\fn\fn\fn\fn\f}\fn\fn\fn\fn\fn\fn\fn\fn\fn\fnh}\f}\c}\f {2+{\sqrt {2}}{2}\cdot {\frac {\sqrt} {\c}} {\c}}} {\cdot {\cdot {\fn}\cdot {\fnh}}}}}} {\cdot {\cdot {\cdot {\f}\cdot {\fn\c}}}}}}}}} {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\c}}}}}}}}}}}}\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\cdot {\ {2+{\sqrt {2+{\sqrt {2}}}} {2}\cdot \cdots

Una fórmula de producto doblemente infinito que involucra la secuencia de Thue-Morse:

{\displaystyle {\frac {\fn\fnMicrosoft {\fnMicrosoft {\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\fnMicrosoft {\\\\\\fn\\\\\\\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\\\\\\\\\\\\fn\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\fnMin {2}}=\prod _{m\geq 1}\prod _{n\geq 1}\left({\frac {4m^{2}+n-2)(4m^{2}+2n-1)}{2}{4(2m^{2}+n-1)(4m^{2}+n-1) ♪♪

Donde

\epsilon _{n}=(-1)^{t_{n}

T_{n

es la secuencia Thue-Morse (Tóth 2020).

Fórmulas argentinas

{\frac {\fn\fnMicrosoft {\fnMicrosoft {\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\fnMicrosoft {\\\\\\fn\\\\\\\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\\\\\\\\\\\\fn\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\fnMin {2} {k+1}}=\arctan {\fnMicroc} {2-a_{k-1}} {a_{k}}}\qquad \qquad k\geq 2

{\frac {\fn\fnMicrosoft {\fnMicrosoft {\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\fnMicrosoft {\\\\\\fn\\\\\\\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\\\\\\\\\\\\fn\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\fnMin {4}=\sum _{k\geq 2}\arctan {\fnMicroc} {2-a_{k-1}} {a_{k}}}

Donde $A_{k}={\sqrt {2+a_{k-1}}$ tales que $A_{1}={\sqrt {2}$ .

{\frac {\fn\fnMicrosoft {\fnMicrosoft {\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\fnMicrosoft {\\\\\\fn\\\\\\\\fnMicrosoft {\\\\\\fnMicrosoft {\\\\\\\\\\\\\\\fn\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\fnMin {\fnK} {\fnMicroc {1} {\fn0}}}=\arctan {\fn}}=\arctan [\frac {1}{1}}+\arctan {\frac {1}{2}+\arctan {\frac {1}{5}}+\arctan {\frac {1}{13}+\cdots

Donde $F_{k$ es k- el número Fibonacci.

\pi =\arctan a+\arctan b+\arctan c

siempre $a+b+c=abc$ y $a$ , $b$ , $c$ son números reales positivos (ver Lista de identidades trigonométricas). Un caso especial

\pi =\arctan 1+\arctan 2+\arctan 3.

Funciones complejas

e^{i\pi} }+1=0

(Identidad de Euler)

Las siguientes equivalencias son verdaderas para cualquier complejo $z$ :

e^{z}\in \mathbb {R} \leftrightarrow \Im z\in \pi \mathbb {Z

E^{z}=1\leftrightarrow z\in 2\pi I 'mathbb {Z

También

{\frac}{e^{z}-1}=\lim - ¿Por qué? ¿Qué? {1}{2}\quad z\in \mathbb {C

Supongamos una celosa $\Omega$ se genera por dos períodos de sesiones ${\displaystyle \omega _{1},\omega ¿Qué?$ . Definimos el cuasi-períodos de esta celosa $\eta _{1}=\zeta (z+\omega _{1};\Omega)-\zeta (z;\Omega)$ y $\eta _{2}=\zeta (z+\omega _{2};\Omega)-\zeta (z;\Omega)$ Donde $\zeta$ es la función Weierstrass zeta ( ${\displaystyle \eta ¿Qué?$ y $\eta _{2$ son de hecho independientes de $z$ ). Luego los períodos y cuasi-períodos están relacionados por los Identidad legendaria:

{\displaystyle \eta _{1}\omega - ¿Qué? ###{2}\omega - ¿Qué?

fracciones continuadas

{\displaystyle {\frac {4}{\i} }=1+{\cfrac {2}{2+{\frac} {3^{2}{2+{\cfrac {5^{2}{2+{\cfrac {7^{2}{2+\ddots - Sí.

{\frac {\fnMicrosoft {\\fnMicrosoft {\\\fnMicrosoft {\\\\\\\\\fnMicrosoft {\\\\\\\\\\\\\\fn\\\\\\\\\\fnMicrosoft {\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ^{2}{\pi }={2+{\cfrac {1}{4+{\frac} {3^{2}{4+{\cfrac {5^{2}{4+{\cfrac {7^{2} {4+\ddots}}}}}\quad

(Ramanujan,

\varpi

es la constante de lemniscate)

\pi ={3+{\cfrac {1}{6+{\frac {3^{2}{6+{\cfrac {5^{2}{6+{\cfrac {7^{2}{6+\ddots {\fnMicrosoft Sans Serif}}}}

{\displaystyle \pi ={\cfrac {4}{1+{\cfrac {1^{2}{3+{\cfrac {2}{5+{\cfrac {3^{2}{7+{\cfrac {4^{2}{9+\ddots - Sí.

{\displaystyle 2\pi ={6+{\cfrac {2}{12+{\cfrac {6^{2}{12+{\cfrac {10}{2}{12+{\cfrac {14}{2}{12+{\cfrac {18^{2}{12+\ddots - Sí.

{\displaystyle \pi =4-{\cfrac {2}{1+{\cfrac {1}{1-{\cfrac {1}{1+{2}{1-{\cfrac {2}{1+{\cfrac {3}{1-{\cfrac {3}{\ddots - Sí.

Para obtener más información sobre la cuarta identidad, consulte la fórmula de fracción continua de Euler.

(Véase también Fracción continua y Fracción continua generalizada.)

algoritmos iterativos

a_{0}=1,\,a_{n+1}=\left(1+{\frac {1}{2n+1}\right)a_{n},\pi =\lim _{n\to \infty }{\frac {\fn}} {\fn}}} {\fn}} {\fn}}} {\fn}}}}} {\n}}}}}}} {\n} {\fn}}}} {}}}}}}}} {}}}}} {}}}}} {}}}}}} {}}}}}}}} {}}}}}}} {}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}} {}}}}}}}}}}} {}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}

a_{1}=0,\,a_{n+1}={\sqrt {2+a_{n}}\,\pi =\lim _{n\to \infty }2^{n}{n}{\sqrt {2-a_{n}}

(cercamente relacionado con la fórmula de Viète)

\omega (i_{n},i_{n-1},\dotsi_{1})=2+i_{n}{\sqrt {2+i_{n-1}{\sqrt {2+\cdots +i_{1}{\sqrt {2}}}=\omega (b_{n},b_{n-1},\dotsb_{1}),\,i_{k}\in {-1,1\},\,b_{\begin{cases}0 limit{if} }i_{k}=1\1 }i_{k}=-1\end{cases}\,\pi ={\displaystyle \lim _{n\rightarrow \infty }{\frac {2^{n+1}{2h+1}}{\sqrt {\omega\left}}}

(donde)

g_{m,h+1

es la entrada h+1-th del código gris m-bit,

h\in \left\{0,1,\ldots2^{m}-1\right\

)

\forall k\in \mathbb {N}\,a_{1}=2^{-k},\,a_{n+1}=a_{n}+2^{-k}(1-\tan(2^{k-1}a_{n})),\,\pi =2^{k+1}\lim _{n\to \infty }a_{n

(convergencia cuadrada)

a_{1}=1,\,a_{n+1}=a_{n}+\sin a_{n},\,\pi =\lim _{n\to \infty }a_{n

(convergencia cúbica)

A_{0}=2{\sqrt {3},\,b_{0}=3,\,a_{n+1}=\operatorname {hm} (a_{n},b_{n}),\,b_{n+1}=\operatorname {gm} (a_{n+1},b_{n}),\,\pi =\lim _{n\to \infty }a_{n}=\lim _{n\to \infty }b_{n

(El algoritmo de Arquimedes, ver también la media armónica y la media geométrica)

Para conocer algoritmos más iterativos, consulte el algoritmo de Gauss-Legendre y el algoritmo de Borwein.

Asintotics

{\fn} {\fn}}\sim {\fn} {\fn} {\fn} {\fn} {\fn}}} {\fn}}} {\fn}}}} {\fn}}} {\fn}}}} {\fn}}}}}}}}}}}}}}}

(tasa de crecimiento asintotico de los coeficientes binomiales centrales)

¿Qué? {\pi n^{3}}}

(tasa de crecimiento sintomático de los números catalanes)

n!\sim {2\sqrt}\left({\frac {\n}}\right)} {\n}} {\n}}}} {\n}}}

(Su aproximación)

\log n!\simeq \left(n+{\frac {1}\right)\log n-n+{\frac {\log 2\pi } {2}

\sum _{k=1}\varphi (k)\sim {\frac {3n^{2}{\pi ^{2}

(donde)

\varphi

es la función totiente de Euler)

\sum _{k=1}{n}{\frac {\varphi (k)}{k}\sim {\frac} {6n}{\pi ^{2}}

El símbolo $\sim$ significa que ratio del lado izquierdo y el lado derecho tiende a uno como $n\to \infty$ .

El símbolo $\simeq$ significa que diferencia entre el lado izquierdo y el lado derecho tiende a cero como $n\to \infty$ .

Inversiones hipergeométricas

Con ${}_{2}F_{1$ siendo la función hipergeométrica:

\sum _{n=0}{\infty }r_{2}(n)q^{n}={2}F_{1}\left({\frac] {1}{2}},{\frac {1}{2}},1,z\right)

donde

q=\exp \left(-\pi {\frac {{}_{2}F_{1}(1/2,1/2,1/2,1,1-z)}{}{2}F_{1}(1/2,1/2,1,z)}}\right),\quad z\in \mathbb {C} \setminus \{0,1\

y $R_{2$ es la suma de dos cuadrados función.

De manera similar,

1+240\sum _{n=1}{\infty }\sigma _{3}(n)q^{n}={2}F_{1}\left({\frac] {1}{6},{\frac {5}{6},1,z\right)}{4

donde

q=\exp \left(-2\pi {\frac {}_{2}F_{1}(1/6,5/6,1,1-z)}{}_{2}F_{1}(1/6,5/6,1,z)}}\right),\quad z\in \mathbb {C} \setminus \{0,1\

y ${\displaystyle \sigma ¿Qué?$ es una función divisor.

Se pueden dar más fórmulas de esta naturaleza, como lo explica la teoría de Ramanujan de funciones elípticas para bases alternativas.

Tal vez las inversiones hipergeométricas más notables son los dos ejemplos siguientes, con la función de Ramanujan tau $\tau$ y los coeficientes Fourier $\mathrm {j$ de la J-invarianteOEIS: A000521):

\sum _{n=-1}{\infty }\mathrm {j} _{n}q^{n}=256{\dfrac {(1-z+z^{2}{3}{z^{2}(1-z)}}{2}}}}

\sum _{n=1}{\infty }\tau (n)q^{n}={\dfrac {2}(1-z)}{256}{2} {}_{2}F_{1}\left({\frac} {1}{2},{\frac {1}{2}},1,z\right)} {12}

donde en ambos casos

q=\exp \left(-2\pi {\frac {{}_{2}F_{1}(1/2,1/2,1/2,1,1-z)}{}{2}F_{1}(1/2,1/2,1,z)}}\right),\quad z\in \mathbb {C} \setminus \{0,1\}.

Además, al expandir la última expresión como una serie de potencias en

{\dfrac {1}{2}{\dfrac {1-(1-z)^{1/4}{1+(1-z)^{1/4}}

y configuración $z=1/2$ , obtenemos una serie rápidamente convergente para $e^{-2\pi$ :

e^{-2\pi }=w^{2}+4w^{6}+34w^{10}+360w^{14}+4239w^{18}+\cdots\quad w={\dfrac {1}{2}{\dfrac {2^{1/4}-1}{2^{1/4}+1}

Varios

\Gamma (s)\Gamma (1-s)={\frac {\pi } {\sin \pi s}

(La fórmula de reflexión de Euler, vea la función Gamma)

\pi ^{-s/2}\Gamma \left({\frac {}{2}\right)\zeta (s)=\pi ^{-(1-s)/2}\Gamma \left({\frac] {1-s}{2}\derecha)\zeta (1-s)

(la ecuación funcional de la función Riemann zeta)

e^{-\zeta}={\sqrt {2\pi}

e^{\zeta '(0,1/2)-\zeta '(0,1)}={\sqrt {\pi

(donde)

\zeta (s,a)

es la función Hurwitz zeta y el derivado se toma con respecto a la primera variable)

\pi =\mathrm {B} (1/2,1/2)=\Gamma (1/2)^{2

(véase también la función Beta)

\pi ={\frac {\Gamma (3/4)^{4}} {\fnMimbre {agm} (1,1/{\sqrt {2}}}}}={\frac {\Gamma\left({1/4}\right)^{4/3}\operatorname {agm} {\fnMicrosoft Sans}} {2}}} {2}}

(donde el agm es la media aritmética-geométrica)

{\displaystyle \pi =\operatorname {agm} \left(\theta _{2}^{2}(1/e),\theta _{3}^{2}(1/e)\right

(donde)

\theta _{2

{\displaystyle \theta ¿Qué?

son las funciones de Jacobi theta)

\operatorname {agm} (1,{\sqrt {2}}={\frac {\pi ♫{\varpi

(debido a Gauss,

\varpi

es la constante de lemniscate)

\operatorname {N} (2\varpi)=e^{2\pi},\quad \operatorname {N} (\varpi)=e^{\pi /2

(donde)

\operatorname {N

es la función de Gauss N)

i\pi =\operatorname {Log} (-1)=\lim _{n\to\infty }n\left((-1)^{1/n}-1\right)

(donde)

\operatorname {Log

es el valor principal del logaritmo complejo)

{\displaystyle 1-{\frac} ♫ {2}{12}=\lim ¿Por qué? {1}{n^{2}}\sum ¿Qué?

(donde)

{\fnK}

es el resto en la división de n por k)

\pi =\lim _{r\to \infty}{\frac {1}{r^{2}}}\sum - ¿Qué? {\fnMicrosoft Sans Serif} } {\ sqrt {x^{2}+y^{2}}\leq . . . . . . . {x^{2}+y^{2}} {\fnMicrosoft Sans Serif}

(Suponiendo el área de un círculo)

\pi =\lim _{n\rightarrow \infty}{\frac {4}{n^{2}}\sum ¿Qué? {n^{2}-k^{2}}

(Riemann suma para evaluar el área del círculo de unidad)

\pi =\lim _{n\to \infty}{\frac {2}=\lim _{n\rightarrow \infty }{2n}{2n}{2n\choose ### {2}}=\lim _{n\rightarrow \infty }{\frac {1}{n}}\left({\frac {(2n)!}{(2n-1)}\right)}{2}}}}}}}} {2}}}}}}}}}}}}}} {2}}}} {}}}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}

( combinando la aproximación de Stirling con el producto Wallis)

\pi =\lim _{n\to \infty}{\frac {1}{n}\ln {\frac {16}{\lambda (ni)}} {\fn} {\fn}} {\fn} {\fn}} {\fn\fn\fn\fnK\fnK}}}}} {\f}}}}}}}}}}}}}}}}}}}}}} {\f}}}}}}} {\f}}}}}}}}}

(donde)

\lambda

es la función modular de lambda)

\pi =\lim _{n\to \infty}{\frac {24}{\sqrt {n}}\ln \left(2^{1/4}G_{n}\right)=\lim _{n\toinfty }{24}{\sqrt {\n}} {\ln}{\n} {\n}}}{\n}{\n}}}}}}}{\n}{\n}{}}}}}}{\n}{\n}{\n}}}{}}}}}{}{\n}}}}}}{}}}}}}}}}}}}{}{}}}}{}{}{}}}}{}}}}}}}{}}}}}{}{}{}}}}}}}}}}}}}}}}}}}}}}{}}}}}}}{}}}}}}}}}}}}}}}}}}

(donde)

G_{n

g_{n

son invariantes de clase de Ramanujan)

Véase también

Lista de identidades matemáticas
Listas de temas de matemáticas
Lista de identidades trigonométricas – Igualdad que implica funciones trigonométricas
Lista de temas relacionados con π – Temas relacionados con la constante matemática
Lista de representaciones de e

Referencias

Notas

^ La relación $\mu _{0}=4\cdot 10^{-7}\,\mathrm {N} /\mathrm {A} ^{2$ fue válido hasta la redefinición 2019 de las unidades base SI.
^ (forma integrada de arctan sobre todo su dominio, dando el período de tan)
^ Los coeficientes se pueden obtener revirtiendo la serie Puiseux de
${\displaystyle z\mapsto {\fn\fn} {\fn\fn\fn\fn} {\fn\fn\fn} {\fn\fn} {\fn\fn} {\fn\fn\fn}\fn\fn\fn} {\fn\fn}}}}}}}}}}}}}}}}}} {\f}}}}}}}}}}}}}}}}}}}} {\f}}}}}}}}}}}}}} {\f}}}}} {\f}}}}}}}}}}} {\f}}}}}}}}}}} {\f}}}} {\f}}}}}}} {\f}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {\$
a $z=0$ .
^ El $n$ th root with the smallest positive principal argument is chosen.
^ Cuando $n\in \mathbb {\fnK$ , esto da aproximaciones algebraicas a la constante de Gelfond $e^{\pi}$ .
^ Cuando ${\sqrt {n}\en \mathbb {\fnK$ , esto da aproximaciones algebraicas a la constante de Gelfond $e^{\pi}$ .

Otros

^ Galperin, G. (2003). "Jugar piscina con π (el número π de un punto de vista billar)" (PDF). Dinámica regular y caótica. 8 (4): 375–394. doi:10.1070/RD2003v008n04ABEH000252.
^ Rudin, Walter (1987). Análisis real y complejo (Tercera edición). McGraw-Hill Book Company. ISBN 0-07-100276-6. p. 4
^ A000796 – OEIS
^ Carson, B. C. (2010), "Elliptic Integrals", en Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), Manual NIST de Funciones Matemáticas, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
^ Arndt, Jörg; Haenel, Christoph (2001). π Unleashed. Springer-Verlag Berlin Heidelberg. ISBN 978-3-540-66572-4. página 126
^ Gourdon, Xavier. "Computación del n-th dígito decimal de π con memoria baja" (PDF). Números, constantes y computación. P. 1.
^ Weisstein, Eric W. "Pi Formulas", MathWorld
^ Chrystal, G. (1900). Álgebra, un libro de texto elemental: Parte IIPág. 335.
^ Eymard, Pierre; Lafon, Jean-Pierre (2004). El número Pi. American Mathematical Society. ISBN 0-8218-3246-8. p. 112
^ Cooper, Shaun (2017). Funciones Theta de Ramanujan (Primera edición). Springer. ISBN 978-319-56171-4. (página 647)
^ Euler, Leonhard (1748). Introductio in analysin infinitorum (en latín). Vol. 1. p. 245
^ Carl B. Boyer, Una historia de matemáticas, Capítulo 21., págs. 488 a 489
^ Euler, Leonhard (1748). Introductio in analysin infinitorum (en latín). Vol. 1. p. 244
^ Wästlund, Johan. "Resumiendo cuadrados inversos por geometría euclidiana" (PDF). El papel da la fórmula con un signo menos, pero estos resultados son equivalentes.
^ Simon Plouffe / David Bailey. "El mundo de Pi". Pi314.net. Retrieved 2011-01-29.
"Colección de serie para π". Numbers.computation.free.fr. Retrieved 2011-01-29.
^ Rudin, Walter (1987). Análisis real y complejo (Tercera edición). McGraw-Hill Book Company. ISBN 0-07-100276-6. p. 3
^ a b Loya, Paul (2017). Aspectos de análisis asombrosos y estéticos. Springer. p. 589. ISBN 978-1-4939-6793-3.
^ Perron, Oskar (1957). Die Lehre von den Kettenbrüchen: Band II (en alemán) (Tercera edición). B. G. Teubner. p. 36, eq. 24
^ Vellucci, Pierluigi; Bersani, Alberto Maria (2019-12-01). "$\pi $$-Formulas y código gris". Ricerche di Matematica. 68 (2): 551-569. arXiv:1606.09597. doi:10.1007/s11587-018-0426-4. ISSN 1827-3491. S2CID 119578297.
^ Abrarov, Sanjar M.; Siddiqui, Rehan; Jagpal, Rajinder K.; Quine, Brendan M. (2021-09-04). " Determinación algorítmica de un gran entero en la fórmula de doble mecanizado tipo π". Matemáticas. 9 (17): 2162. arXiv:2107.01027. doi:10.3390/math9172162.
^ Arndt, Jörg; Haenel, Christoph (2001). π Unleashed. Springer-Verlag Berlin Heidelberg. ISBN 978-3-540-66572-4. página 49
^ Eymard, Pierre; Lafon, Jean-Pierre (2004). El número Pi. American Mathematical Society. ISBN 0-8218-3246-8. p. 2
^ Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi y la AGM: Un estudio en la teoría del número analítico y la complejidad computacional (Primera edición). Wiley-Interscience. ISBN 0-471-83138-7. página 225
^ Gilmore, Tomack. "El Medio Aritmético-Geométrico de Gauss" (PDF). Universität Wien. Pág. 13.
^ Borwein, J.; Borwein, P. (2000). "Ramanujan y Pi". Pi: Un libro fuente. Springer Link. pp. 588–595. doi:10.1007/978-1-4757-3240-5_62. ISBN 978-1-4757-3242-9.
^ Eymard, Pierre; Lafon, Jean-Pierre (2004). El número Pi. American Mathematical Society. ISBN 0-8218-3246-8. p. 248

Tóth, László (2020), "Productos Infinitos Transcendentales Asociados con el +-1 Thue-Morse Sequence" (PDF), Journal of Integer Sequences, 23: 20.8.2, arXiv:2009.02025.

Más lectura

Borwein, Peter (2000). "El número increíble π" (PDF). Nieuw Archief voor Wiskunde5a serie. 1 (3): 254–258. ZBL 1173.01300.
Kazuya Kato, Nobushige Kurokawa, Saito Takeshi: Teoría Número 1: Sueño de Fermat. American Mathematical Society, Providence 1993, ISBN 0-8218-0863-X.

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