# Pi

## Definitions

Symbol: Pi $\pi$ The constant pi (3.14...)

## Numerical value

$\pi \in \left[3.1415926535897932384626433832795028841971693993751 \pm 5.83 \cdot 10^{-51}\right]$
Table of simple expressions involving $\pi$ to 50 digits
$\pi \notin \mathbb{Q}$
$\pi \notin \overline{\mathbb{Q}}$
$\pi = \sum_{n=1}^{\infty} \text{A000796}\!\left(n\right) {10}^{1 - n}$

## Euler's identity

${e}^{\pi i} + 1 = 0$

## Elementary function representations

$\pi = 4 \operatorname{atan}(1)$
$\pi = 2 \operatorname{acos}(0)$
$\pi = 2 \operatorname{asin}(1)$
$\pi = \mathop{\operatorname{zero*}\,}\limits_{x \in \left[3, 4\right]} \sin(x)$
$\pi = -i \log(-1)$
$\pi = 10 \operatorname{asin}\!\left(\frac{1}{2 \varphi}\right)$

### Machin-type formulas

$\pi = 16 \operatorname{atan}\!\left(\frac{1}{5}\right) - 4 \operatorname{atan}\!\left(\frac{1}{239}\right)$
$\pi = 4 \operatorname{atan}\!\left(\frac{1}{2}\right) + 4 \operatorname{atan}\!\left(\frac{1}{3}\right)$
$\pi = 8 \operatorname{atan}\!\left(\frac{1}{2}\right) - 4 \operatorname{atan}\!\left(\frac{1}{7}\right)$
$\pi = 8 \operatorname{atan}\!\left(\frac{1}{3}\right) + 4 \operatorname{atan}\!\left(\frac{1}{7}\right)$
$\pi = 4 \operatorname{atan}\!\left(\frac{1}{2}\right) + 4 \operatorname{atan}\!\left(\frac{1}{5}\right) + 4 \operatorname{atan}\!\left(\frac{1}{8}\right)$
$\pi = 4 \operatorname{atan}\!\left(\frac{1}{3}\right) + 4 \operatorname{atan}\!\left(\frac{1}{4}\right) + 4 \operatorname{atan}\!\left(\frac{1}{7}\right) + 4 \operatorname{atan}\!\left(\frac{1}{13}\right)$
$\pi = 48 \operatorname{atan}\!\left(\frac{1}{49}\right) + 128 \operatorname{atan}\!\left(\frac{1}{57}\right) - 20 \operatorname{atan}\!\left(\frac{1}{239}\right) + 48 \operatorname{atan}\!\left(\frac{1}{110443}\right)$

## Integral representations

$\pi = 2 \int_{-1}^{1} \sqrt{1 - {x}^{2}} \, dx$
$\pi = \int_{-1}^{1} \frac{1}{\sqrt{1 - {x}^{2}}} \, dx$
$\pi = \int_{-\infty}^{\infty} \frac{1}{{x}^{2} + 1} \, dx$
$\pi = {\left(\int_{-\infty}^{\infty} {e}^{-{x}^{2}} \, dx\right)}^{2}$
$\pi = \frac{22}{7} - \int_{0}^{1} \frac{{x}^{4} {\left(1 - x\right)}^{4}}{1 + {x}^{2}} \, dx$
$\pi = \frac{355}{113} - \frac{1}{3164} \int_{0}^{1} \frac{{x}^{8} {\left(1 - x\right)}^{8} \left(25 + 816 {x}^{2}\right)}{1 + {x}^{2}} \, dx$
$\pi = \int_{-\infty}^{\infty} \operatorname{sinc}(x) \, dx$
$\pi = \int_{-\infty}^{\infty} \operatorname{sinc}^{2}\!\left(x\right) \, dx$
$\pi = 2 e \int_{0}^{\infty} \frac{\cos(x)}{{x}^{2} + 1} \, dx$
$\pi = 8 {\left(\int_{0}^{\infty} \sin\!\left({x}^{2}\right) \, dx\right)}^{2}$
$\pi = 8 {\left(\int_{0}^{\infty} \cos\!\left({x}^{2}\right) \, dx\right)}^{2}$
$\pi = \int_{0}^{\infty} \theta_{2}\!\left(0 , i t\right) \, dt$
$\pi = 3 \int_{0}^{\infty} \left(\theta_{3}\!\left(0 , i t\right) - 1\right) \, dt$

## Series representations

$\pi = 4 \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{2 n + 1}$
$\pi = \sum_{n=0}^{\infty} \frac{{2}^{n} {\left(n !\right)}^{2}}{\left(2 n + 1\right)!}$
$\pi = \sum_{n=0}^{\infty} \frac{n !}{\left(2 n + 1\right)!!}$
$\pi = \sum_{n=0}^{\infty} \frac{1}{{16}^{n}} \left(\frac{4}{8 n + 1} - \frac{2}{8 n + 4} - \frac{1}{8 n + 5} - \frac{1}{8 n + 6}\right)$
$\frac{1}{\pi} = \frac{2 \sqrt{2}}{9801} \sum_{n=0}^{\infty} \frac{\left(4 n\right)! \left(1103 + 26390 n\right)}{{\left(n !\right)}^{4} \cdot {396}^{4 n}}$
$\frac{1}{\pi} = 12 \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n} \left(6 n\right)! \left(13591409 + 545140134 n\right)}{\left(3 n\right)! {\left(n !\right)}^{3} \cdot {640320}^{3 n + 3 / 2}}$
$\pi = 72 \sum_{n=1}^{\infty} \frac{1}{n \left({e}^{\pi n} - 1\right)} - 96 \sum_{n=1}^{\infty} \frac{1}{n \left({e}^{2 \pi n} - 1\right)} + 24 \sum_{n=1}^{\infty} \frac{1}{n \left({e}^{4 \pi n} - 1\right)}$
$\pi = 8 \sum_{n=0}^{\infty} \frac{1}{\left(4 n + 1\right) \left(4 n + 3\right)}$
$\frac{{\pi}^{2}}{6} = \sum_{n=1}^{\infty} \frac{1}{{n}^{2}}$
$\frac{{\pi}^{2}}{12} = \sum_{n=1}^{\infty} \frac{{\left(-1\right)}^{n + 1}}{{n}^{2}}$
$\frac{{\pi}^{4}}{90} = \sum_{n=1}^{\infty} \frac{1}{{n}^{4}}$
$\pi = 2 \sum_{n=0}^{\infty} \operatorname{atan}\!\left(\frac{1}{{n}^{2} + n + 1}\right)$
$\pi = 2 \sum_{n=0}^{\infty} \operatorname{atan}\!\left(\frac{1}{F_{2 n + 1}}\right)$
$\pi = \sqrt{3} \left(3 \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{3 n + 1} - \log(2)\right)$
$\pi = 4 \sqrt{2} \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{4 n + 1} - 2 \log\!\left(1 + \sqrt{2}\right)$
$\pi = \sqrt{3} \left(\frac{9}{2} \sum_{n=0}^{\infty} \frac{1}{{2 n \choose n}} - 6\right)$
$\pi = \sqrt{3} \left(\frac{9}{2} \sum_{n=1}^{\infty} \frac{n}{{2 n \choose n}} - 3\right)$
$\pi = \sum_{n=1}^{\infty} \frac{\left({3}^{n} - 1\right) \zeta\!\left(n + 1\right)}{{4}^{n}}$

## Product representations

$\pi = 2 \prod_{n=1}^{\infty} \frac{4 {n}^{2}}{4 {n}^{2} - 1}$
$\pi = 2 \prod_{n=2}^{\infty} \sec\!\left(\frac{\pi}{{2}^{n}}\right)$
$\frac{{\pi}^{2}}{6} = \prod_{p} {\left(1 - \frac{1}{{p}^{2}}\right)}^{-1}$
$\frac{2}{\pi} = \prod_{n=1}^{\infty} \frac{a_{n}}{2}\; \text{ where } a_{1} = \sqrt{2},\;a_{n} = \sqrt{2 + a_{n - 1}}$

## Limit representations

$\pi = \lim_{n \to \infty} \frac{4}{{n}^{2}} \sum_{k=0}^{n} \sqrt{{n}^{2} - {k}^{2}}$
$\pi = \lim_{n \to \infty} \frac{{16}^{n}}{n {{2 n \choose n}}^{2}}$
$\pi = \lim_{n \to \infty} \frac{1}{2} {\left({\left(-1\right)}^{n + 1} \frac{\left(2 n\right)!}{B_{2 n}}\right)}^{1 / \left(2 n\right)}$
$\frac{3}{{\pi}^{2}} = \lim_{N \to \infty} \frac{1}{{N}^{2}} \sum_{n=1}^{N} \varphi(n)$
$\pi = \frac{4 {\left(\operatorname{agm}\!\left(1, \frac{1}{\sqrt{2}}\right)\right)}^{2}}{1 - \sum_{j=0}^{\infty} {2}^{j} c_{j}^{2}} = \lim_{n \to \infty} \frac{{\left(a_{n} + b_{n}\right)}^{2}}{1 - \sum_{j=0}^{n} {2}^{j} c_{j}^{2}}\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(1, \frac{1}{\sqrt{2}}\right),\;c_{n} = a_{n} - b_{n}$

## Special function representations

$\pi = {\left(\Gamma\!\left(\frac{1}{2}\right)\right)}^{2}$
$\pi = \frac{\sqrt{3}}{2} \Gamma\!\left(\frac{1}{3}\right) \Gamma\!\left(\frac{2}{3}\right)$
$\pi = \frac{1}{\sqrt{2}} \Gamma\!\left(\frac{1}{4}\right) \Gamma\!\left(\frac{3}{4}\right)$
$\pi = \sqrt{6 \zeta\!\left(2\right)}$
$\pi = \mathrm{B}\!\left(\frac{1}{2}, \frac{1}{2}\right)$
$\pi = G_{2}\!\left(i\right)$
$\pi = \frac{1}{2} {\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{4 / 3} {\left(\operatorname{agm}\!\left(1, \sqrt{2}\right)\right)}^{2 / 3}$
$\pi = 2 K(0)$
$\pi = 2 E(0)$
$\pi = \sqrt{6 \operatorname{Li}_{2}\!\left(1\right)}$
$\pi = 2 \,{}_2F_1\!\left(\frac{1}{2}, \frac{1}{2}, \frac{3}{2}, 1\right)$
$\frac{1}{\pi} = \frac{1}{2} \,{}_2F_1\!\left(\frac{1}{2}, -\frac{1}{2}, 1, 1\right)$
$\frac{1}{\pi} = \frac{1}{4} \,{}_2F_1\!\left(-\frac{1}{2}, -\frac{1}{2}, 1, 1\right)$
$\pi = 2 \,{}_2F_1\!\left(-\frac{1}{2}, -\frac{1}{2}, \frac{1}{2}, 1\right)$
$\pi = 4 \left(\,{}_2F_1\!\left(-\frac{1}{2}, 1, \frac{1}{2}, -1\right) - 1\right)$
$\pi = 2 \,{}_2F_1\!\left(1, 1, \frac{1}{2}, \frac{1}{2}\right) - 4$
$\pi = 4 \left(\sqrt{2} \,{}_2F_1\!\left(-\frac{1}{2}, -\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right) - 1\right)$
$\pi = \sqrt{3} \left(\frac{9}{2} \,{}_2F_1\!\left(1, 1, \frac{1}{2}, \frac{1}{4}\right) - 6\right)$
$\pi = 12 \,{}_2F_1\!\left(-\frac{1}{2}, -\frac{1}{2}, \frac{1}{2}, \frac{1}{4}\right) - 6 \sqrt{3}$
$\pi = \frac{9}{2 \sqrt{3}} \,{}_2F_1\!\left(1, 1, \frac{3}{2}, \frac{1}{4}\right)$
$\frac{1}{\pi} = \frac{2 \sqrt{3}}{9} \,{}_2F_1\!\left(-\frac{1}{3}, \frac{1}{3}, 1, 1\right)$
$\pi = \frac{5 \sqrt{\varphi + 2}}{2 \varphi} \,{}_2F_1\!\left(1, 1, \frac{3}{2}, \frac{1}{{\left(2 \varphi\right)}^{2}}\right)$
$\pi = \sqrt{\psi'\!\left(\frac{1}{4}\right) - 8 G}$
$\pi = 4 L\!\left(1, \chi_{4 \, . \, 3}\right)$

## Approximations

$\left|\pi - \frac{22}{7}\right| < 0.00127$
$\left|\pi - \frac{355}{113}\right| < 2.67 \cdot 10^{-7}$
$\left|\pi - \frac{\log\!\left({640320}^{3} + 744\right)}{\sqrt{163}}\right| < 2.24 \cdot 10^{-31}$
$\left|\frac{1}{\pi} - \left(12 \sum_{n=0}^{N - 1} \frac{{\left(-1\right)}^{n} \left(6 n\right)! \left(13591409 + 545140134 n\right)}{\left(3 n\right)! {\left(n !\right)}^{3} \cdot {640320}^{3 n + 3 / 2}}\right)\right| < \frac{1}{{151931373056000}^{N}}$
$\left|\pi - \frac{{\left(a_{n} + b_{n}\right)}^{2}}{1 - \sum_{j=0}^{n} {2}^{j} c_{j}^{2}}\right| \le {2}^{n + 8} {e}^{-\pi {2}^{n + 1}}\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(1, \frac{1}{\sqrt{2}}\right),\;c_{n} = a_{n} - b_{n}$

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-04-08 16:14:44.404316 UTC