List of formulae involving π

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The following is a list of formulae involving the mathematical constant π:

1 Classical geometry
2 Analysis
3 Physics
4 References

[edit] Classical geometry

$C = 2\pi r,\,$

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

$A = \pi r^2,\,$

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

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

$\int_{-1}^1\frac{dx}{\sqrt{1-x^2}} = \pi$

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

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

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

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

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

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

[edit] Other infinite series

$\sum_{n=0}^{\infty} \frac{(-1)^{n}}{2n+1} = \frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots = \arctan{1} = \frac{\pi}{4}$

$\zeta(2) = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots = \frac{\pi^2}{6}$ (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)

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