List of formulae involving π

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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.

1 Classical geometry
2 Analysis
3 Physics
4 References
5 See also

[edit] 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.

[edit] Analysis

[edit] Integrals

$\int_{-1}^1 \sqrt{1-x^2}\,dx = \frac{\pi}{2}$ (see π)

$\int_{-1}^1\frac{dx}{\sqrt{1-x^2}} = \pi$ (see π)

$\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}$ (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)

$\int_0^\infty\frac{\sin(x)}{x}\,dx=\frac{\pi}{2}.$

$\int_0^1 {x^4(1-x)^4 \over 1+x^2}\,dx = {22 \over 7} - \pi$ (see also proof that 22 over 7 exceeds π).

[edit] Efficient infinite series

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

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

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

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

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

[edit] 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)

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

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

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

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

[edit] Machin-like formulae

[edit] 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:

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