List of formulae involving π

Part of a series of articles on
the mathematical constant π
Uses
Area of disk Circumference Use in other formulae
Properties
Irrationality Transcendence Less than 22/7
Value
Approximations Memorization
People
Archimedes Liu Hui Zu Chongzhi Madhava of Sangamagrama William Jones John Machin John Wrench Ludolph van Ceulen Aryabhata
History
Chronology Book
In culture
Legislation Holiday
Related topics
Squaring the circle Basel problem Tau (τ) Other topics related to 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 on pi, or the one on numerical approximations of pi.

Classical geometry

$C = 2 \pi r = \pi d\!$

where C is the circumference of a circle, r is the radius and 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.

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

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

Analysis

Integrals

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

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

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

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

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

$\oint\frac{dz}{z}=2\pi i\!$ (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)

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

The following are good 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)

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

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

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

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

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

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

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

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

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

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

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

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 2 or 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.

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)

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

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

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

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

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

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

Infinite products

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

$e^{i \pi} = -1\!$ (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)

$\lim_{n\rightarrow \infty} 10^{n+2}\cdot \sin(\frac{1}{\underbrace{55\cdots5}_{\mathrm{n\; digits}}}) = \pi\!$ (where the sine function is in degrees, not radians)

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:

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

• 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

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

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.

See also

• Pi
• Approximations of π
• List of topics related to π
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