List of formulae involving π explained

The following is a list of significant formulae involving the mathematical constant . Many of these formulae can be found in the article Pi, or the article Approximations of .

Euclidean geometry

\pi = (Cd)/(=) (C{ 2 r})/() where is the circumference of a circle, is the diameter, and is the radius. More generally,\pi = (L)/(w) where and are, respectively, the perimeter and the width of any curve of constant width.A = \pi r ^{2} where is the area of a circle. More generally,A = \pi ab where is the area enclosed by an ellipse with semi-major axis and semi-minor axis .C = (2 \pi)/(\operatorname{agm) (a,b)}\left(a ^{inf ty} _{n =2} 2 ^{n-1} (a ^{2 )\right)} _{n} where is the circumference of an ellipse with semi-major axis and semi-minor axis and a _{n,b} _{n} are the arithmetic and geometric iterations of \operatorname{agm}(a,b) , the arithmetic-geometric mean of and with the initial values a _{0=a} and b _{0=b} .A =4 \pi r ^{2} where is the area between the witch of Agnesi and its asymptotic line; is the radius of the defining circle.A = (\Gamma ( 1 / 4 ) ^{2})/(2 \sqrt{\pi) } r^2=\fracwhere is the area of a squircle with minor radius, \Gamma is the gamma function.A = (k +1 )(k +2 )\pi r ^{2} where is the area of an epicycloid with the smaller circle of radius and the larger circle of radius (k\in N ), assuming the initial point lies on the larger circle.A = (( -1 ) ^{k+3})/(8) \pi a ^{2} where is the area of a rose with angular frequency (k\in N ) and amplitude .L = (\Gamma ( 1 / 4 ) ^{2})/(\sqrt{\pi) }c=\fracwhere is the perimeter of the lemniscate of Bernoulli with focal distance .V = { 4 \over 3 }\pi r ^{3} where is the volume of a sphere and is the radius.SA = 4 \pi r ^{2} where is the surface area of a sphere and is the radius.H = { 1 \over 2 }\pi ^{2} r ^{4} where is the hypervolume of a 3-sphere and is the radius.SV = 2 \pi ^{2} r ^{3} where is the surface volume of a 3-sphere and is the radius.

Regular convex polygons

Sum of internal angles of a regular convex polygon with sides:

S = (n -2 )\pi Area of a regular convex polygon with sides and side length :A = (ns ^{2})/(\cot 4) (\pi)/(n) Inradius of a regular convex polygon with sides and side length :r = (s)/(\cot 2) (\pi)/(n) Circumradius of a regular convex polygon with sides and side length :R = (s)/(\csc 2) (\pi)/(n)

Physics

• The cosmological constant:

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

• Heisenberg's uncertainty principle

\Delta x \Delta p \ge (h)/(4 \pi)

• Einstein's field equation of general relativity:

R _{\mu\nu} - (1)/(2) g _{\mu\nu} R + Λ g _{\mu\nu} = { 8 \pi G \over c ^{4}} T _{\mu\nu}

• Coulomb's law for the electric force in vacuum:

F = (|q _{1q} _{2|})/(4 \pi \varepsilon _{0} r ^{2})

• Magnetic permeability of free space

^[1]

\mu _{0} ≈ 4 \pi ⋅ 10 ^{-7} N / A ^{2}

• Approximate period of a simple pendulum with small amplitude:

T ≈ 2 \pi \sqrt (L)/(g)

• Exact period of a simple pendulum with amplitude

\theta _{0} (\operatorname{agm} is the arithmetic–geometric mean):T = (2 \pi)/(\operatorname{agm) ( 1 ,\cos (\theta _{0 / 2 ))}\sqrt{ (L)/(g)} }

• Kepler's third law of planetary motion

(R ^{3})/(T ^{2}) = (GM)/(4 \pi ^{2})

• The buckling formula:

F = (\pi ^{2EI})/(L ^{2})

A puzzle involving "colliding billiard balls":

\lfloor{b ^{N\pi}\rfloor} is the number of collisions made (in ideal conditions, perfectly elastic with no friction) by an object of mass m initially at rest between a fixed wall and another object of mass b^2Nm, when struck by the other object.^[2] (This gives the digits of π in base b up to N digits past the radix point.)

Formulae yielding

Integrals

2 \int ^{1} _{-1} \sqrt{ 1- x ^{2}dx} = \pi (integrating two halves y(x) = \sqrt{ 1- x ^{2}} to obtain the area of the unit circle)\int ^{inf ty} _{-inf ty} \operatorname{sech}x dx = \pi \int ^{inf ty} _{-inf ty} \int ^{inf ty} _{t} e ^{-1 / 2 t ^{2-x} ^{2+xt}} dx dt = \int ^{inf ty} _{-inf ty} \int ^{inf ty} _{t} e ^{- t ^{2-1/2x} ^{2+xt}} dx dt = \pi \int ^{1 (dx)/(\sqrt{ 1- x ^{2})} _{-1} } = \pi\int ^{inf ty (dx)/(1+ x ^{2})} _{-inf ty} = \pi ^{[3] [4]} (see also Cauchy distribution)\int ^{inf ty} _{-inf ty} (\sin x)/(x) dx = \pi (see Dirichlet integral)\int ^{inf ty} _{-inf ty} e ^{- x ^{2}} dx = \sqrt{\pi} (see Gaussian integral).\oint (dz)/(z) = 2 \pi i (when the path of integration winds once counterclockwise around 0. See also Cauchy's integral formula).\int ^{inf ty} _{0} ln \left( 1+ (1)/(x ^{2}) \right) dx = \pi ^[5] \int ^{inf ty} _{-inf ty} (\sin x)/(x) dx = \pi \int ^{1} _{0} {x ^{4(1-x)} ^{4} \over 1+ x ^{2}dx} = { 22 \over 7 } - \pi (see also Proof that 22/7 exceeds ).\int ^{1} _{0} {x ^{2(1+x)} ^{4} \over 1+ x ^{2}dx} = \pi - { 17 \over 15 } \int ^{inf ty} _{0} (x ^{\alpha-1})/(x +1) dx = (\pi)/(\sin \pi\alpha) , 0 <\alpha< 1 \int ^{inf ty} _{0} (dx)/(\sqrt{x(x + a)(x + b)) }=\frac (where \operatorname{agm} is the arithmetic–geometric mean; see also elliptic integral)

Note that with symmetric integrands

f( - x) = f(x) , formulas of the form $\backslash int\_^af(x)\backslash ,dx$ can also be translated to formulas $2\backslash int\_^af(x)\backslash ,dx$.

Efficient infinite series

\sum ^{inf ty} _{k =0} (k!)/(( 2 k +1 )!!) = \sum ^{inf ty (2 ^{k} k! ^{2})/(( 2 k +1 )!)} _{k =0} = (\pi)/(2) (see also Double factorial)\sum ^{inf ty} _{k =0} (k!)/(= 2 ^{k(2k+1)!!}) (2 \pi)/(3 \sqrt{ 3) }\sum ^{inf ty} _{k =0} (k!( 2 k)!( 25 k -3 ))/(= ( 3 k)! 2 ^{k}) (\pi)/(2) \sum ^{inf ty} _{k =0} (( -1 ) ^{k} ( 6 k)! ( 13591409+545140134 k))/(= ( 3 k)!(k!) ^{3} 640320 ^{3k}) (4270934400)/(\sqrt{ 10005) \pi} (see Chudnovsky algorithm)\sum ^{inf ty} _{k =0} (( 4 k)!( 1103+26390 k))/(= (k!) ^{4} 396 ^{4k}) (9801)/(2 \sqrt{ 2) \pi} (see Srinivasa Ramanujan, Ramanujan–Sato series)

The following are efficient for calculating arbitrary binary digits of :

\sum ^{inf ty} _{k =0} (( -1 ) ^{k})/(\left( 4 ^{k}) (2)/(+ 4 k +1) (2)/(+ 4 k +2) (1)/(4 k +3) \right) = \pi ^[6] \sum ^{inf ty} _{k =0} (1)/(16 ^{k}) \left( (4)/(8 k +1) - (2)/(8 k +4) - (1)/(8 k +5) - (1)/(8 k +6) \right) = \pi (see Bailey–Borwein–Plouffe formula)\sum ^{inf ty} _{k =0} (1)/(\left( 16 ^{k}) (8)/(+ 8 k +2) (4)/(+ 8 k +3) (4)/(- 8 k +4) (1)/(8 k +7) \right) =2 \pi \sum ^{inf ty} _{k =0} ({( -1 ))/(^{k}{2}) ^{10k} } \left(- \frac - \frac + \frac - \frac - \frac - \frac + \frac \right)=2^6\pi

Plouffe's series for calculating arbitrary decimal digits of :^[7]

\sum ^{inf ty} _{k =1} k (2 ^{kk!} ^{2})/(( 2 k)!) = \pi +3

Other infinite series

\zeta( 2 ) = (1)/(1 ^{2}) + (1)/(2 ^{2}) + (1)/(3 ^{2}) + (1)/(4 ^{2}) + … = (\pi ^{2})/(6) (see also Basel problem and Riemann zeta function)\zeta( 4 ) = (1)/(1 ^{4}) + (1)/(2 ^{4}) + (1)/(3 ^{4}) + (1)/(4 ^{4}) + … = (\pi ^{4})/(90) \zeta( 2 n) = \sum ^{inf ty} _{k =1} (1)/(k ^{2n}) = (1)/(1 ^{2n}) + (1)/(2 ^{2n}) + (1)/(3 ^{2n}) + (1)/(4 ^{2n}) + … = ( -1 ) ^{n+1} (B _{2n} ( 2 \pi) ^{2n})/(2 ( 2 n)!) , where B_2n is a Bernoulli number.\sum ^{inf ty} _{n =1} (3 ^{n} -1)/(4 ^{n}) \zeta(n +1 ) = \pi ^[8] \sum ^{inf ty} _{n =1} (7 ^{n} -1)/(8 ^{n}) \zeta(n +1 ) = ( 1+ \sqrt{ 2 })\pi \sum ^{inf ty} _{n =2} (2 ( 3 / 2 ) ^{n-3})/(n) (\zeta (n) -1 ) =ln \pi \sum ^{inf ty} _{n =1} \zeta ( 2 n) (x ^{2n})/(=ln n) (\pi x)/(\sin \pi x) , 0 <|x|< 1 \sum ^{inf ty} _{n =0} (( -1 ) ^{n})/(2 n +1) = 1 - (1)/(3) + (1)/(5) - (1)/(7) + (1)/(9) - … = \arctan{ 1 } = (\pi)/(4) (see Leibniz formula for pi)\sum ^{inf ty} _{n =0} (( -1 ) ^{(n ^{2-n)/2}})/(=1+ 2 n +1) (13- (15- (17+ (19+ (1)/(11))/(- … = (\pi)/(2 \sqrt{ 2)))/())/())/() } (Newton, Second Letter to Oldenburg, 1676)^[9] \sum ^{inf ty} _{n =0} (( -1 ) ^{n})/(=1- 3 ^{n(2n+1)}) (1)/(+ 3 ^{1 ⋅} 3) (1)/(- 3 ^{2 ⋅} 5) (1)/(+ 3 ^{3 ⋅} 7) (1)/(3 ^{4 ⋅} 9) - … = \sqrt{ 3 }\arctan (1)/(\sqrt{ 3) }=\frac (Madhava series)\sum ^{inf ty} _{n =1} (( -1 ) ^{n+1})/(= n ^{2}) (1)/(1 ^{2}) - (1)/(2 ^{2}) + (1)/(3 ^{2}) - (1)/(4 ^{2}) + … = (\pi ^{2})/(12) \sum ^{inf ty} _{n =1} (1 {( 2 n))/(^{2}}) = (1)/(2 ^{2}) + (1)/(4 ^{2}) + (1)/(6 ^{2}) + (1)/(8 ^{2}) + … = (\pi ^{2})/(24) \sum ^{inf ty} _{n =0} \left( (1)/(2 n +1) \right) ^{2} = (1)/(1 ^{2}) + (1)/(3 ^{2}) + (1)/(5 ^{2}) + (1)/(7 ^{2}) + … = (\pi ^{2})/(8) \sum ^{inf ty} _{n =0} \left( (( -1 ) ^{n})/(2 n +1) \right) ^{3} = (1)/(1 ^{3}) - (1)/(3 ^{3}) + (1)/(5 ^{3}) - (1)/(7 ^{3}) + … = (\pi ^{3})/(32) \sum ^{inf ty} _{n =0} \left( (1)/(2 n +1) \right) ^{4} = (1)/(1 ^{4}) + (1)/(3 ^{4}) + (1)/(5 ^{4}) + (1)/(7 ^{4}) + … = (\pi ^{4})/(96) \sum ^{inf ty} _{n =0} \left( (( -1 ) ^{n})/(2 n +1) \right) ^{5} = (1)/(1 ^{5}) - (1)/(3 ^{5}) + (1)/(5 ^{5}) - (1)/(7 ^{5}) + … = (5 \pi ^{5})/(1536) \sum ^{inf ty} _{n =0} \left( (1)/(2 n +1) \right) ^{6} = (1)/(1 ^{6}) + (1)/(3 ^{6}) + (1)/(5 ^{6}) + (1)/(7 ^{6}) + … = (\pi ^{6})/(960)

In general,

\sum ^{inf ty} _{n =0} (( -1 ) ^{n})/(( 2 n +1 ) ^{2k+1}) = ( -1 ) ^{k} (E _{2k})/(\left( 2 ( 2 k)!) (\pi)/(2) \right) ^{2k+1} , k\in N _{0}

where

E _{2k} is the 2 k th Euler number.^[10] \sum ^{inf ty} _{n =0} \binom{ (1)/(2) }\frac = 1 - \frac - \frac-\cdots = \frac\sum ^{inf ty} _{n =0} (1)/(( 4 n +1 )( 4 n +3 )) = (1)/(+ 1 ⋅ 3) (1)/(+ 5 ⋅ 7) (1)/(+ … = 9 ⋅ 11) (\pi)/(8) \sum ^{inf ty} _{n =1} ( -1 ) ^{(n ^{2+n)/2+1}} \left|G _{\left(( -1 ) ^{n+1} +6 n -3 \right)/ 4} \right| = |G _{1|+|G} _{2|-|G} _{4|-|G} _{5|+|G} _{7|+|G} _{8|-|G} _{10} | - |G _{11} | + … = (\sqrt{ 3)/() } (see Gregory coefficients)\sum ^{inf ty} _{n =0} (( 1 / 2 ) ^{2} _{n})/(2 ^{n} n! ^{2}) \sum ^{inf ty} _{n =0} (n( 1 / 2 ) ^{2} _{n})/(= 2 ^{n} n! ^{2}) (1)/(\pi) (where (x) _{n} is the rising factorial)^[11] \sum ^{inf ty} _{n =1} (( -1 ) ^{n+1})/(n(n +1 )( 2 n +1 )) = \pi -3 (Nilakantha series)\sum ^{inf ty} _{n =1} (F _{2n})/({n}} = n ^{2} \binom{ 2 n) (4 \pi ^{2})/(25 \sqrt 5) (where F _{n} is the n-th Fibonacci number)\sum ^{inf ty} _{n =1} \sigma(n)e ^{-2\pi} = (1)/(- 24) (1)/(8 \pi) (where \sigma is the sum-of-divisors function)\pi = \sum ^{inf ty} _{n =1} (( -1 ) ^{\epsilon})/(n) =1 + (1)/(2) + (1)/(3) + (1)/(4) - (1)/(5) + (1)/(6) + (1)/(7) + (1)/(8) + (1)/(9) - (1)/(10) + (1)/(11) + (1)/(12) - (1)/(13) + …   (where \epsilon (n) is the number of prime factors of the form p\equiv 1 ( mod 4 ) of n )^{[12] [13]} (\pi)/(2) = \sum ^{inf ty} _{n =1} (( -1 ) ^{\varepsilon})/(=1+ n) (1)/(- 2) (1)/(+ 3) (1)/(+ 4) (1)/(- 5) (1)/(- 6) (1)/(+ 7) (1)/(+ 8) (1)/(9) + …   (where \varepsilon (n) is the number of prime factors of the form p\equiv 3 ( mod 4 ) of n )^[14] \pi = \sum ^{inf ty} _{n =-inf ty} (( -1 ) ^{n})/(n +1 / 2) \pi ^{inf ty} _{n =-inf ty} (1)/((n +1 / 2 ) ^{2}) ^[15]

The last two formulas are special cases of

\begin{align} (\pi)/(\sin\pi x) & = \sum ^{inf ty} _{n =-inf ty} (( -1 ) ^{n})/(\\ \left( n + x) (\pi)/(\sin \pi x) \right) ^{inf ty} _{n =-inf ty} (1)/((n + x) ^{2}) \end{align}

which generate infinitely many analogous formulas for

\pi when x\in Q \setminus Z .

Some formulas relating and harmonic numbers are given here. Further infinite series involving π are:^[16]

\pi = (1)/(Z)	Z = \sum ^{inf ty} _{n =0} ((( 2 n)!) ^{3(42n+5)})/((n!) ^{6{16}) ^{3n+1} }
\pi = (4)/(Z)	Z = \sum ^{inf ty} _{n =0} (( -1 ) ^{n(4n)!(21460n+1123)})/((n!) ^{4{441}) ^{2n+1} { 2 } ^{10n+1} }
\pi = (4)/(Z)	Z = \sum ^{inf ty} _{n =0} (( 6 n +1 )\left ( (1)/(2) \right ) ^{3} _{n})/({ 4 ^{n}) (n!) ^{3}}
\pi = (32)/(Z)	Z = \sum ^{inf ty} _{n =0} \left ( (\sqrt{ 5)/(-1 }{ 2 }) \right ) ^{8n} (( 42 n\sqrt{ 5)/(+30 n) + 5 \sqrt{ 5 } -1 ) \left ( (1)/(2) \right ) ^{3} _{n}} {{ 64 ^{n}(n!)} ^{3}}
\pi = (27)/(4 Z)	Z = \sum ^{inf ty} _{n =0} \left ( (2)/(27) \right ) ^{n} (( 15 n +2 )\left ( (1)/(2) \right ) _{n} \left ( (1)/(3) \right ) _{n} \left ( (2)/(3) \right ) _{n})/((n!) ^{3})
\pi = (15 \sqrt{ 3)/() }	Z = \sum ^{inf ty} _{n =0} \left ( (4)/(125) \right ) ^{n} (( 33 n +4 )\left ( (1)/(2) \right ) _{n} \left ( (1)/(3) \right ) _{n} \left ( (2)/(3) \right ) _{n})/((n!) ^{3})
\pi = (85 \sqrt{ 85)/() }	Z = \sum ^{inf ty} _{n =0} \left ( (4)/(85) \right ) ^{n} (( 133 n +8 )\left ( (1)/(2) \right ) _{n} \left ( (1)/(6) \right ) _{n} \left ( (5)/(6) \right ) _{n})/((n!) ^{3})
\pi = (5 \sqrt{ 5)/() }	Z = \sum ^{inf ty} _{n =0} \left ( (4)/(125) \right ) ^{n} (( 11 n +1 )\left ( (1)/(2) \right ) _{n} \left ( (1)/(6) \right ) _{n} \left ( (5)/(6) \right ) _{n})/((n!) ^{3})
\pi = (2 \sqrt{ 3)/() }	Z = \sum ^{inf ty} _{n =0} (( 8 n +1 )\left ( (1)/(2) \right ) _{n} \left ( (1)/(4) \right ) _{n} \left ( (3)/(4) \right ) _{n})/((n!) ^{3{9}) ^{n} }
\pi = (\sqrt{ 3)/() }	Z = \sum ^{inf ty} _{n =0} (( 40 n +3 )\left ( (1)/(2) \right ) _{n} \left ( (1)/(4) \right ) _{n} \left ( (3)/(4) \right ) _{n})/((n!) ^{3{49}) ^{2n+1} }
\pi = (2 \sqrt{ 11)/() }	Z = \sum ^{inf ty} _{n =0} (( 280 n +19 )\left ( (1)/(2) \right ) _{n} \left ( (1)/(4) \right ) _{n} \left ( (3)/(4) \right ) _{n})/((n!) ^{3{99}) ^{2n+1} }
\pi = (\sqrt{ 2)/() }	Z = \sum ^{inf ty} _{n =0} (( 10 n +1 ) \left ( (1)/(2) \right ) _{n} \left ( (1)/(4) \right ) _{n} \left ( (3)/(4) \right ) _{n})/((n!) ^{3{9}) ^{2n+1} }
\pi = (4 \sqrt{ 5)/() }	Z = \sum ^{inf ty} _{n =0} (( 644 n +41 ) \left ( (1)/(2) \right ) _{n} \left ( (1)/(4) \right ) _{n} \left ( (3)/(4) \right ) _{n})/((n!) ^{35} ^{n{72}) ^{2n+1} }
\pi = (4 \sqrt{ 3)/() }	Z = \sum ^{inf ty} _{n =0} (( -1 ) ^{n( 28 n +3 )} \left ( (1)/(2) \right ) _{n} \left ( (1)/(4) \right ) _{n} \left ( (3)/(4) \right ) _{n})/((n!) ^{3{3} ^{n}) { 4 } ^{n+1} }
\pi = (4)/(Z)	Z = \sum ^{inf ty} _{n =0} (( -1 ) ^{n( 20 n +3 )} \left ( (1)/(2) \right ) _{n} \left ( (1)/(4) \right ) _{n} \left ( (3)/(4) \right ) _{n})/((n!) ^{3{2}) ^{2n+1} }
\pi = (72)/(Z)	Z = \sum ^{inf ty} _{n =0} (( -1 ) ^{n(4n)!(260n+23)})/((n!) ^{44} ^{4n} 18 ^{2n})
\pi = (3528)/(Z)	Z = \sum ^{inf ty} _{n =0} (( -1 ) ^{n(4n)!(21460n+1123)})/((n!) ^{44} ^{4n} 882 ^{2n})

where

(x) _{n} is the Pochhammer symbol for the rising factorial. See also Ramanujan–Sato series.

Machin-like formulae

See also: Machin-like formula.

(\pi)/(4) = \arctan 1 (\pi)/(4) = \arctan (1)/(2) + \arctan (1)/(3) (\pi)/(4) = 2 \arctan (1)/(2) - \arctan (1)/(7) (\pi)/(4) = 2 \arctan (1)/(3) + \arctan (1)/(7) (\pi)/(4) = 4 \arctan (1)/(5) - \arctan (1)/(239) (the original Machin's formula)(\pi)/(4) = 5 \arctan (1)/(7) + 2 \arctan (3)/(79) (\pi)/(4) = 6 \arctan (1)/(8) + 2 \arctan (1)/(57) + \arctan (1)/(239) (\pi)/(4) = 12 \arctan (1)/(49) + 32 \arctan (1)/(57) - 5 \arctan (1)/(239) + 12 \arctan (1)/(110443) (\pi)/(4) = 44 \arctan (1)/(57) + 7 \arctan (1)/(239) - 12 \arctan (1)/(682) + 24 \arctan (1)/(12943)

Infinite products

(\pi)/(4) = \left(\prod _{p\equiv} (p)/(p -1) \right) ⋅ \left( \prod _{p\equiv} (p)/(\right) = p +1) (3)/(4) ⋅ (5)/(4) ⋅ (7)/(8) ⋅ (11)/(12) ⋅ (13)/(12) … , (Euler)

where the numerators are the odd primes; each denominator is the multiple of four nearest to the numerator.

(\sqrt{ 3)/(\pi}{ 6 } = \left(\displaystyle\prod) _{p} \atop p \in P } (p)/(p -1) \right) ⋅ \left(\displaystyle\prod _{p} \atop p \in P } (p)/(\right) = p +1) (5)/(6) ⋅ (7)/(6) ⋅ (11)/(12) ⋅ (13)/(12) ⋅ (17)/(18) … (\pi)/(2) = \prod ^{inf ty} _{n =1} (( 2 n)( 2 n))/(( 2 n -1 )( 2 n +1 )) = (2)/(1) ⋅ (2)/(3) ⋅ (4)/(3) ⋅ (4)/(5) ⋅ (6)/(5) ⋅ (6)/(7) ⋅ (8)/(7) ⋅ (8)/(9) … (see also Wallis product)(\pi)/(2) = \prod ^{inf ty\left( 1+ (1)/(n)} _{n =1} \right) ^{( -1 ) ^{n+1}} = \left( 1+ (1)/(1) \right) ^{+1} \left( 1+ (1)/(2) \right) ^{-1} \left( 1+ (1)/(3) \right) ^{+1} … (another form of Wallis product)

Viète's formula

(2)/(= \pi) (\sqrt 2)/(2) ⋅ (\sqrt{ 2+ \sqrt 2)/() }2 \cdot \frac2 \cdot \cdots

A double infinite product formula involving the Thue–Morse sequence:

(\pi)/(2) = \prod _{m\geq1} \prod _{n\geq1} \left( (( 4 m ^{2} + n -2 ) ( 4 m ^{2} +2 n -1 ) ^{2})/(4 ( 2 m ^{2} + n -1 ) ( 4 m ^{2} + n -1 ) ( 2 m ^{2} + n)) \right) ^{\epsilon _{n}} ,

where

\epsilon _{n} = ( -1 ) ^{t _{n}} and t _{n} is the Thue–Morse sequence .

Arctangent formulas

(\pi)/(2 ^{k+1}) = \arctan (\sqrt{ 2- a _{k-1})/() }, \qquad\qquad k\geq 2 (\pi)/(4) = \sum _{k\geq} \arctan (\sqrt{ 2- a _{k-1})/() }, where a _{k=\sqrt{2+a} _{k-1} } such that a _{1=\sqrt{2}} .(\pi)/(2) = \sum ^{inf ty} _{k =0} \arctan (1)/(F _{2k+1}) = \arctan (1)/(1) + \arctan (1)/(2) + \arctan (1)/(5) + \arctan (1)/(13) + …

where

F _{k} is the k-th Fibonacci number.\pi = \arctan a + \arctan b + \arctan c whenever a + b + c = abc and a , b , c are positive real numbers (see List of trigonometric identities). A special case is\pi = \arctan 1+ \arctan 2+ \arctan 3 .

Complex functions

e ^{i} +1 = 0 (Euler's identity)z :e ^{z\inR\leftrightarrow\Im} z\in\pi Z e ^{z=1\leftrightarrow} z\in 2 \pi i Z ^[17] Also(1)/(e ^{z-1}) = \lim _{N\toinfty} \sum ^{N} _{n =- N} (1)/(- z -2 \pi i n) (1)/(2) , z\in C . \Omega is generated by two periods \omega _{1,\omega} _{2} . We define the quasi-periods of this lattice by η _{1=\zeta} (z + \omega _{1;\Omega)-\zeta} (z;\Omega) and η _{2=\zeta} (z + \omega _{2;\Omega)-\zeta} (z;\Omega) where \zeta is the Weierstrass zeta function (η _{1} and η _{2} are in fact independent of z ). Then the periods and quasi-periods are related by the Legendre identity:η _{1\omega} _{2-η} _{2\omega} _{1=2\pi} i.

Continued fractions

(4)/(\pi) = 1 + \cfrac{ 1 ^{2}{2} + \cfrac{ 3 ^{2}{2} + \cfrac{ 5 ^{2}{2} + \cfrac{ 7 ^{2}{2} + \ddots}}}} ^[18] (\varpi ^{2})/(\pi) = { 2 + \cfrac{ 1 ^{2}{4} + \cfrac{ 3 ^{2}{4} + \cfrac{ 5 ^{2}{4} + \cfrac{ 7 ^{2}{4} + \ddots}}}}} (Ramanujan, \varpi is the lemniscate constant)^[19] \pi = { 3 + \cfrac{ 1 ^{2}{6} + \cfrac{ 3 ^{2}{6} + \cfrac{ 5 ^{2}{6} + \cfrac{ 7 ^{2}{6} + \ddots}}}}} ^[18] \pi = \cfrac{ 4 }{ 1 + \cfrac{ 1 ^{2}{3} + \cfrac{ 2 ^{2}{5} + \cfrac{ 3 ^{2}{7} + \cfrac{ 4 ^{2}{9} + \ddots}}}}} 2 \pi = { 6 + \cfrac{ 2 ^{2}{12} + \cfrac{ 6 ^{2}{12} + \cfrac{ 10 ^{2}{12+} \cfrac{ 14 ^{2}{12} + \cfrac{ 18 ^{2}{12} + \ddots}}}}}} \pi =4- \cfrac{ 2 }{ 1+ \cfrac{ 1 }{ 1- \cfrac{ 1 }{ 1+ \cfrac{ 2 }{ 1- \cfrac{ 2 }{ 1+ \cfrac{ 3 }{ 1- \cfrac{ 3 }{\ddots}}}}}}} For more on the fourth identity, see Euler's continued fraction formula.

(See also Continued fraction and Generalized continued fraction.)

Iterative algorithms

a _{0=1,} a _{n+1} = \left( 1+ (1)/(2 n +1) \right)a _{n,} \pi = \lim _{n\toinfty} (a ^{2} _{n})/(n) a _{1=0,} a _{n+1} = \sqrt{ 2+ a _{n},} \pi = \lim _{n\toinfty} 2 ^{n\sqrt{ 2- a} _{n}} (closely related to Viète's formula)\omega(i _{n,i} _{n-1} , ... ,i _{1} ) =2+ i _{n} \sqrt{ 2+ i _{n-1} \sqrt{ 2+ … + i _{1} \sqrt{ 2 }}} = \omega(b _{n,b} _{n-1} , ... ,b _{1} ), i _{k} \in\{ -1 , 1 \}, b _{k=\begin{cases}} 0 & if i _{k=1\\} 1 & if i _{k=-1} \end{cases}, \pi = {\displaystyle\lim _{n} (2 ^{n+1})/(2 h +1) \sqrt{\omega\left(\underbrace{ 10 \ldots 0 } _{n-m} g _{m,} \right)}} (where g _{m,} is the h+1-th entry of m-bit Gray code, h \in \left\{ 0 , 1 , \ldots, 2 ^{m} -1 \right\} )^[20] \forall k \in N , a _{1} = 2 , a _{n} = a _{n} + 2 ( 1 - \tan ( 2 ^{k} a _{n)),} \pi = 2 ^{k} \lim _{n} a _{n} (quadratic convergence)^[21] a _{1=1,} a _{n+1} = a _{n+\sin} a _{n,} \pi = \lim _{n\toinfty} a _{n} (cubic convergence)^[22] 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\toinfty} a _{n=\lim} _{n\toinfty} b _{n} (Archimedes' algorithm, see also harmonic mean and geometric mean)^[23]

For more iterative algorithms, see the Gauss–Legendre algorithm and Borwein's algorithm.

Asymptotics

\binom{ 2 n}{n}\sim (4 ^{n})/(\sqrt{\pi n) } (asymptotic growth rate of the central binomial coefficients)C _{n\sim} (4 ^{n})/(\sqrt{\pi n ^{3}) } (asymptotic growth rate of the Catalan numbers)n! \sim \sqrt{ 2 \pi n} \left( (n)/(e) \right) ^{n} (Stirling's approximation)log n!\simeq \left(n + (12 \right) log)/(n - n + (log2 \pi)/(2)) \sum ^{n} _{k =1} \varphi (k) \sim (3 n ^{2})/(\pi ^{2}) (where \varphi is Euler's totient function)\sum ^{n} _{k =1} (\varphi (k))/(k) \sim (6 n)/(\pi ^{2})

The symbol

\sim means that the ratio of the left-hand side and the right-hand side tends to one as n\to inf ty .

The symbol

\simeq means that the difference between the left-hand side and the right-hand side tends to zero as n\to inf ty .

Hypergeometric inversions

With

{} _{2F} _{1} being the hypergeometric function:\sum ^{inf ty} _{n =0} r ^{n = {}} _{2 F} ^{1 \left( (12 , (12 , 1 ,z\right))/())/()} whereq = \exp\left( - \pi ({)/(_{2F}) _{1(1/2,1/2,1,1-z)}{{}} _{2F} _{1(1/2,1/2,1,z)}\right),} z\in C \setminus\{ 0 , 1 \} and r _{2} is the sum of two squares function.

Similarly,

1+240 \sum ^{inf ty} _{n =1} \sigma ^{n} _{3 (n)q} = {} _{2F} ^{1 \left( (16 , (56 , 1 ,z\right))/(^{4}))/()} whereq = \exp\left( -2 \pi ({)/(_{2F}) _{1(1/6,5/6,1,1-z)}{{}} _{2F} _{1(1/6,5/6,1,z)}\right),} z\in C \setminus\{ 0 , 1 \} and \sigma _{3} is a divisor function.

More formulas of this nature can be given, as explained by Ramanujan's theory of elliptic functions to alternative bases.

\tau and the Fourier coefficients j of the J-invariant :\sum ^{inf ty} _{n =-1} j ^{n =256 \dfrac{( 1- z + z} _{nq} ^{2)} ^{3}{z} ^{2(1-z)} ^{2},} \sum ^{inf ty} _{n =1} \tau(n)q ^{n=\dfrac{z} ^{2(1-z)} ^{2 }{ 256 }{}} ^{2 F} ^{1 \left( (12 , (12 , 1 ,z\right))/(^{12}))/()} where in both casesq = \exp\left( -2 \pi ({)/(_{2F}) _{1(1/2,1/2,1,1-z)}{{}} _{2F} _{1(1/2,1/2,1,z)}\right),} z\in C \setminus\{ 0 , 1 \}.

Furthermore, by expanding the last expression as a power series in

\dfrac{ 1 }{ 2 }\dfrac{ 1- ( 1- z) ^{1/4} }and setting z =1 / 2 , we obtain a rapidly convergent series for e ^{-2\pi} :^[24] e ^{-2\pi} = w ^{2+4w} ^{6+34w} ^{10} +360 w ^{14} +4239 w ^{18} + … , w = \dfrac{ 1 }{ 2 }\dfrac{ 2 ^{1/4} -1 }{ 2 ^{1/4} +1 }.

Miscellaneous

\Gamma (s)\Gamma ( 1- s) = (\pi)/(\sin \pi s) (Euler's reflection formula, see Gamma function)\pi ^{-s/2} \Gamma \left( (s)/(2) \right)\zeta (s) = \pi ^{-(1-s)/2} \Gamma\left( (1- s)/(2) \right)\zeta ( 1- s) (the functional equation of the Riemann zeta function)e ^{-\zeta'(0)} = \sqrt{ 2 \pi} e ^{\zeta'(0,1/2)-\zeta'(0,1)} = \sqrt{\pi} (where \zeta (s,a) is the Hurwitz zeta function and the derivative is taken with respect to the first variable)\pi = \Beta ( 1 / 2 , 1 / 2 ) = \Gamma ( 1 / 2 ) ^{2} (see also Beta function)\pi = (\Gamma ( 3 / 4 ) ^{4})/(\operatorname{agm) ( 1 , 1 /\sqrt{ 2 }) ^{2 } = (\Gamma\left({ 1 / 4)/(\right))} ^{4/3} \operatorname{agm}( 1 , \sqrt{ 2 }) ^{2/3} } (where agm is the arithmetic–geometric mean)\pi = \operatorname{agm}\left(\theta ^{2 ( 1 /e)\right)} _{3} (where \theta _{2} and \theta _{3} are the Jacobi theta functions^[25])\operatorname{agm}( 1 ,\sqrt{ 2 }) = (\pi)/(\varpi) (due to Gauss,^[26] \varpi is the lemniscate constant)\operatorname{N}( 2 \varpi) = e ^{2\pi} , \operatorname{N}(\varpi) = e ^{\pi/2} (where \operatorname{N} is the Gauss N-function)i\pi = \operatorname{Log}( -1 ) = \lim _{n\toinfty} n\left(( -1 ) ^{1/n} -1 \right) (where \operatorname{Log} is the principal value of the complex logarithm)^[27] 1- (\pi ^{2})/(12) = \lim _{n →} (1)/(n ^{2}) \sum ^{n} _{k =1} (n\bmod k) (where $n\backslash bmod\; k$ is the remainder upon division of n by k)\pi = \lim _{r} (1)/(r ^{2}) \sum ^{r} _{x =- r} \sum ^{r} _{y =- r} \begin{cases} 1 & if \sqrt{x ^{2+y} ^{2}} \le r \\ 0 & if \sqrt{x ^{2+y} ^{2}} > r \end{cases} (summing a circle's area)\pi = \lim _{n} (4)/(n ^{2}) \sum ^{n} _{k =1} \sqrt{n ^{2} - k ^{2}} (Riemann sum to evaluate the area of the unit circle)\pi = \lim _{n\toinfty} (2 ^{4n} n! ^{4})/(n( 2 n)! ^{2}) = \lim _{n} (2 ^{4n})/(n { 2 n\choose n) ^{2}} = \lim _{n} (1)/(\left( n) (( 2 n)!!)/(( 2 n -1 )!!) \right) ^{2} (by combining Stirling's approximation with Wallis product)\pi = \lim _{n\toinfty} (1)/(ln n) (16)/(λ (ni)) (where λ is the modular lambda function)^{[28] [29]} \pi = \lim _{n\toinfty} (24)/(\sqrt{n) }\ln \left(2^ G_n\right)=\lim_\frac\ln \left(2^g_n\right) (where G _{n} and g _{n} are Ramanujan's class invariants)^{[30] [31]}

References

Other

• .

Further reading

• Borwein . Peter . 3 . Nieuw Archief voor Wiskunde . 254–258 . 5th series . The amazing number . 1 . 2000 . 1173.01300.
• Kazuya Kato, Nobushige Kurokawa, Saito Takeshi: Number Theory 1: Fermat's Dream. American Mathematical Society, Providence 1993, .

Notes and References

1. The relation \mu _{0} = 4 \pi ⋅ 10 ^{-7} N / A ^{2} was valid until the 2019 redefinition of the SI base units.
2. Galperin . G. . 2003 . Playing pool with π (the number π from a billiard point of view) . Regular and Chaotic Dynamics . 8 . 4 . 375–394. 10.1070/RD2003v008n04ABEH000252 .
3. Book: Rudin . Walter. Walter Rudin. Real and Complex Analysis . McGraw-Hill Book Company . 1987 . Third . 0-07-100276-6. p. 4
4. (integral form of arctan over its entire domain, giving the period of tan)
5. https://oeis.org/A000796 A000796 – OEIS
6. Book: Arndt . Jörg . Haenel. Christoph . π Unleashed . Springer-Verlag Berlin Heidelberg . 2001. 978-3-540-66572-4. page 126
7. Web site: Computation of the n-th decimal digit of π with low memory. Gourdon. Xavier. Numbers, constants and computation. 1.
8. http://mathworld.wolfram.com/PiFormulas.html Weisstein, Eric W. "Pi Formulas", MathWorld
9. Book: Chrystal . G. . Algebra, an Elementary Text-book: Part II . 1900 . 335.
10. Book: Eymard . Pierre . Lafon. Jean-Pierre . The Number Pi . American Mathematical Society . 2004 . 0-8218-3246-8. p. 112
11. Book: Cooper . Shaun . Ramanujan's Theta Functions . Springer . 2017 . First . 978-3-319-56171-4. (page 647)
12. Book: Euler . Leonhard . Introductio in analysin infinitorum . 1. Latin. 1748. p. 245
13. [Carl B. Boyer]
14. Book: Euler . Leonhard . Introductio in analysin infinitorum . 1. Latin. 1748. p. 244
15. Web site: Wästlund . Johan . Summing inverse squares by euclidean geometry. The paper gives the formula with a minus sign instead, but these results are equivalent.
16. Web site: The world of Pi . Simon Plouffe / David Bailey . Pi314.net . 2011-01-29.
    Web site: Collection of series for . Numbers.computation.free.fr . 2011-01-29.
17. Book: Rudin . Walter. Walter Rudin. Real and Complex Analysis . McGraw-Hill Book Company . 1987 . Third . 0-07-100276-6. p. 3
18. Book: Loya . Paul . Amazing and Aesthetic Aspects of Analysis . Springer . 2017 . 978-1-4939-6793-3. 589.
19. Book: Perron . Oskar . German . Oskar Perron. Die Lehre von den Kettenbrüchen: Band II . B. G. Teubner . 1957 . Third. p. 36, eq. 24
20. Vellucci. Pierluigi. Bersani. Alberto Maria. 2019-12-01. $$\pi $$-Formulas and Gray code. Ricerche di Matematica. en. 68. 2. 551–569. 10.1007/s11587-018-0426-4. 1606.09597 . 119578297 . 1827-3491.
21. Abrarov. Sanjar M.. Siddiqui. Rehan. Jagpal. Rajinder K.. Quine. Brendan M.. 2021-09-04. Algorithmic Determination of a Large Integer in the Two-Term Machin-like Formula for π . Mathematics. en. 9. 17. 2162. 10.3390/math9172162 . 2107.01027 . free .
22. Book: Arndt . Jörg . Haenel. Christoph . π Unleashed . Springer-Verlag Berlin Heidelberg . 2001. 978-3-540-66572-4. page 49
23. Book: Eymard . Pierre . Lafon. Jean-Pierre . The Number Pi . American Mathematical Society . 2004 . 0-8218-3246-8. p. 2
24. The coefficients can be obtained by reversing the Puiseux series ofz\mapsto \sqrt{z}\dfrac{\sum ^{inf ty} _{n =0} z ^{2 n ^{2+2n}} }at z =0 .
25. Book: Borwein . Jonathan M. . Borwein. Peter B. . Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity . Wiley-Interscience . 1987 . First . 0-471-83138-7. page 225
26. Web site: The Arithmetic-Geometric Mean of Gauss. Gilmore. Tomack. Universität Wien. 13.
27. The n th root with the smallest positive principal argument is chosen.
28. Book: https://link.springer.com/chapter/10.1007/978-1-4757-3240-5_62. Borwein. J.. Borwein. P.. Pi: A Source Book . Ramanujan and Pi . Springer Link. 2000 . 588–595 . 10.1007/978-1-4757-3240-5_62 . 978-1-4757-3242-9 .
29. When n\in Q ^{+} , this gives algebraic approximations to Gelfond's constant e ^{\pi} .
30. Book: Eymard . Pierre . Lafon. Jean-Pierre . The Number Pi . American Mathematical Society . 2004 . 0-8218-3246-8. p. 248
31. When \sqrt{n}\in Q ^{+} , this gives algebraic approximations to Gelfond's constant e ^{\pi} .