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, or in the articles on π or Computing π.

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.

:$A\; =\; 4pi\; r^2,$

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

Analysis

Integrals

:$int\_\{-1\}^1\; sqrt\{1-x^2\},dx\; =\; frac\{pi\}\{2\}$ (see π)

:$int\_\{-1\}^1frac\{dx\}\{sqrt\{1-x^2\; =\; pi$ (see π)

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

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

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

:$int\_0^inftyfrac\{sin(x)\}\{x\},dx=frac\{pi\}\{2\}.$

:$int\_0^1\; \{x^4(1-x)^4\; over\; 1+x^2\},dx\; =\; \{22\; over\; 7\}\; -\; pi$ (see also proof that 22 over 7 exceeds π).

Efficient infinite series

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

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

The following are good for calculating arbitrary binary digits of π::$sum\_\{k\; =\; 0\}^\{infty\}\; frac\{1\}\{16^k\}\; left(\; frac\{4\}\{8k\; +\; 1\}\; -\; frac\{2\}\{8k\; +\; 4\}\; -\; frac\{1\}\{8k\; +\; 5\}\; -\; frac\{1\}\{8k\; +\; 6\}\; ight)=pi$ (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} ight)=pi

Other infinite series

:$sum\_\{n=0\}^\{infty\}\; frac\{(-1)^\{n\{2n+1\}\; =\; frac\{1\}\{1\}\; -\; frac\{1\}\{3\}\; +\; frac\{1\}\{5\}\; -\; frac\{1\}\{7\}\; +\; frac\{1\}\{9\}\; -\; cdots\; =\; arctan\{1\}\; =\; frac\{pi\}\{4\}$ (see Leibniz formula for pi)

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

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

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

:$sum\_\{n=0\}^\{infty\}\; frac\{1\}\{(2n+1)^2\}\; =\; frac\{1\}\{1^2\}\; +\; frac\{1\}\{3^2\}\; +\; frac\{1\}\{5^2\}\; +\; frac\{1\}\{7^2\}\; +\; cdots\; =\; frac\{pi^2\}\{8\}$

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

Machin-like formulae

See also Machin-like formula.: $frac\{pi\}\{4\}\; =\; 4\; arctanfrac\{1\}\{5\}\; -\; arctanfrac\{1\}\{239\}$ (the original Machin's formula)

:$frac\{pi\}\{4\}\; =\; arctanfrac\{1\}\{2\}\; +\; arctanfrac\{1\}\{3\}$

:$frac\{pi\}\{4\}\; =\; 2\; arctanfrac\{1\}\{2\}\; -\; arctanfrac\{1\}\{7\}$

:$frac\{pi\}\{4\}\; =\; 2\; arctanfrac\{1\}\{3\}\; +\; arctanfrac\{1\}\{7\}$

:$frac\{pi\}\{4\}\; =\; 5\; arctanfrac\{1\}\{7\}\; +\; 2\; arctanfrac\{3\}\{79\}$

:$frac\{pi\}\{4\}\; =\; 12\; arctanfrac\{1\}\{49\}\; +\; 32\; arctanfrac\{1\}\{57\}\; -\; 5\; arctanfrac\{1\}\{239\}\; +\; 12\; arctanfrac\{1\}\{110443\}$

:$frac\{pi\}\{4\}\; =\; 44\; arctanfrac\{1\}\{57\}\; +\; 7\; arctanfrac\{1\}\{239\}\; -\; 12\; arctanfrac\{1\}\{682\}\; +\; 24\; arctanfrac\{1\}\{12943\}$

Infinite products

:$prod\_\{n=1\}^\{infty\}\; frac\{4n^2\}\{4n^2-1\}\; =\; frac\{2\}\{1\}\; cdot\; frac\{2\}\{3\}\; cdot\; frac\{4\}\{3\}\; cdot\; frac\{4\}\{5\}\; cdot\; frac\{6\}\{5\}\; cdot\; frac\{6\}\{7\}\; cdot\; frac\{8\}\{7\}\; cdot\; frac\{8\}\{9\}\; cdots\; =\; frac\{pi\}\{2\}$ (see also Wallis product)

Vieta's formula:

:$frac\{sqrt2\}2\; cdot\; frac\{sqrt\{2+sqrt22\; cdot\; frac\{sqrt\{2+sqrt\{2+sqrt2\}2\; cdot\; cdots\; =\; frac2pi$

Three continued fractions

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

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

:$pi\; =\; cfrac\{4\}\{1\; +\; cfrac\{1\}\{2\; +\; cfrac\{9\}\{2\; +\; cfrac\{25\}\{2\; +\; cfrac\{49\}\{ddots\},$

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

(See also continued fraction and generalized continued fraction.)

Miscellaneous

:$n!\; sim\; sqrt\{2\; pi\; n\}\; left(frac\{n\}\{e\}\; ight)^n$ (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)

:$Gammaleft(\{1\; over\; 2\}\; ight)=sqrt\{pi\}$ (see also gamma function)

:$pi\; =\; frac\{Gammaleft(\{1/4\}\; ight)^\{4/3\}\; mathrm\{agm\}(1,\; sqrt\{2\})^\{2/3\{2\}$ (where agm is the arithmetic-geometric mean)

Physics

*The cosmological constant:::$Lambda\; =$8pi G} over {3c^2 ho
*Heisenberg's uncertainty principle:::$Delta\; x,\; Delta\; p\; ge\; frac\{h\}\{4pi\}$
*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:::$F\; =\; frac\{left|q\_1q\_2\; ight\{4\; pi\; varepsilon\_0\; r^2\}$
*Magnetic permeability of free space:::$mu\_0\; =\; 4\; pi\; cdot\; 10^\{-7\},mathrm\{N/A^2\},$

References

* Peter Borwein, " [http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P159.pdf The Amazing Number Pi] "

ee also

* Pi
* Computing π
* List of topics related to π