List of formulae involving π

The following is a list of significant formulae involving the mathematical constant π. The list contains only formulae whose significance is established either in the article on the formula itself, the article Pi, or the article Approximations of π.

Euclidean Geometry

$\pi = C / d \!$

where C is the circumference of a circle, d is the diameter.

$A = \pi r^2\!$

where A is the area of a circle and r is the radius.

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

where V is the volume of a sphere and r is the radius.

$SA = 4\pi r^2\!$

where SA is the surface area of a sphere and r is the radius.

Physics

• The cosmological constant:

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

• Heisenberg's uncertainty principle:

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

• Einstein's field equation of general relativity:

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

• Coulomb's law for the electric force:

$F = \frac{\left|q_1q_2\right|}{4 \pi \varepsilon_0 r^2}\!$

• Magnetic permeability of free space:

$\mu_0 = 4 \pi \cdot 10^{-7}\,\mathrm{N/A^2}\!$

• Period of a simple pendulum with small amplitude:

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

• The buckling formula:

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

Formulae yielding π

Integrals

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

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

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

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

$\int\limits_{-\infty}^\infty\frac{dx}{1+x^2} = \pi\!$ (integral form of arctan over its entire domain, giving the period of tan).

$\int\limits_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}\!$ (see Gaussian integral).

$\oint\frac{dz}{z}=2\pi i\!$ (when the path of integration winds once counterclockwise around 0. See also Cauchy's integral formula)

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

$\int\limits_0^1 {x^4(1-x)^4 \over 1+x^2}\,dx = {22 \over 7} - \pi\!$ (see also Proof that 22/7 exceeds π).

Efficient infinite series

$\sum_{k=0}^\infty\frac{k!}{(2k+1)!!}=\sum_{k=0}^\infty\frac{2^k k!^2}{(2k+1)!}=\frac{\pi}{2}\!$ (see also Double factorial)

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

$\frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}=\frac{1}{\pi}\!$ (see Srinivasa Ramanujan, Ramanujan–Sato series)

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

The following are efficient for calculating arbitrary binary digits of π:

$\sum_{k = 0}^{\infty} \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6}\right)=\pi\!$ (see Bailey–Borwein–Plouffe formula)

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

Other infinite series

$\zeta(2) = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots = \frac{\pi^2}{6}\!$   (see also Basel problem and Riemann zeta function)

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

$\zeta(2n) = \sum_{k=1}^{\infty} \frac{1}{k^{2n}}\, = \frac{1}{1^{2n}} + \frac{1}{2^{2n}} + \frac{1}{3^{2n}} + \frac{1}{4^{2n}} + \cdots = (-1)^{n+1}\frac{B_{2n}(2\pi)^{2n}}{2(2n)!}\!$ , where B_2n is a Bernoulli number.

$\sum_{n=1}^{\infty} \frac{3^n - 1}{4^n}\, \zeta(n+1) = \pi\!$^[2]

$\sum_{n=0}^{\infty} {\left( \frac{(-1)^{n}}{2n+1} \right) }^1 = \frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots = \arctan{1} = \frac{\pi}{4}\!$   (see Leibniz formula for pi)

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

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

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

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

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

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

After the first two terms, the signs are determined as follows: If the denominator is a prime of the form 4m - 1, the sign is positive; if the denominator is a prime of the form 4m + 1, the sign is negative; for composite numbers, the sign is equal the product of the signs of its factors.^[3]

Also:

$\sum_{n=1}^\infty \frac{F_{2n}}{n^2 \binom{2n}{n}}=\frac{4\pi^2}{25\sqrt5}$

where $F_n$ is the n-th Fibonacci number.

Machin-like formulae

See also Machin-like formula.

$\frac{\pi}{4} = 4 \arctan\frac{1}{5} - \arctan\frac{1}{239} \!$ (the original Machin's formula)

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

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

$\frac{\pi}{4} = 2 \arctan\frac{1}{2} - \arctan\frac{1}{7}\!$

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

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

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

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

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

where $F_n$ is the n'th Fibonacci number.

Infinite series

Some infinite series involving pi are:^[4]

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

where

$(x)_n \!$

is the Pochhammer symbol for the falling factorial. See also Ramanujan–Sato series.

Infinite products

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

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

$\prod_{n=1}^{\infty} \frac{4n^2}{4n^2-1} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \cdots = \frac{4}{3} \cdot \frac{16}{15} \cdot \frac{36}{35} \cdot \frac{64}{63} \cdots = \frac{\pi}{2} \!$ (see also Wallis product)

Vieta's formula:

$\frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdot \cdots = \frac2\pi\!$

Three continued fractions

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

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

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

For more on this third identity, see Euler's continued fraction formula.

(See also Continued fraction and Generalized continued fraction.)

Miscellaneous

$n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n\!$ (Stirling's approximation)

$e^{i \pi} +1 = 0\!$ (Euler's identity)

$\sum_{k=1}^{n} \varphi (k) \sim \frac{3n^2}{\pi^2}\!$ (see Euler's totient function)

$\sum_{k=1}^{n} \frac {\varphi (k)} {k} \sim \frac{6n}{\pi^2}\!$ (see Euler's totient function)

$\Gamma\left({1 \over 2}\right)=\sqrt{\pi}\!$ (see also Gamma function)

$\pi = \frac{\Gamma\left({1/4}\right)^{4/3} \mathrm{agm}(1, \sqrt{2})^{2/3}}{2}\!$ (where agm is the arithmetic-geometric mean)

$\lim_{n\rightarrow \infty}\frac{1}{n^2} \sum_{k=1}^n (n\;\bmod\;k) = 1-\frac{\pi^2}{12}\!$ (where mod is the modulo function which gives the rest of a division this formula is getting better for higher n)

$\pi = \lim_{n \rightarrow \infty} \frac{4}{n^2} \sum_{k=1}^n \sqrt{n^2 - k^2}$ (Riemann sum to evaluate the area of the unit circle)

$\pi = \lim_{n \rightarrow \infty} \frac{2^{4n}}{n {2n\choose n}^2}$ (by Stirling's approximation)

See also

• Pi
• List of topics related to π

References

1. Cetin Hakimoglu-Brown Derivation of Rapidly Converging Infinite Series
2. Weisstein, Eric W. "Pi Formulas", MathWorld
3. Carl B. Boyer, A History of Mathematics, Chapter 21., p. 488-489
4. Simon Plouffe / David Bailey. "The world of Pi". Pi314.net. Retrieved 2011-01-29.
   "Collection of series for π". Numbers.computation.free.fr. Retrieved 2011-01-29.

Further reading

• Peter Borwein, 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.