# Pi

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

## Numerical value

$\pi \in \left[3.1415926535897932384626433832795028841971693993751 \pm 5.83 \cdot 10^{-51}\right]$
$\pi \notin \mathbb{Q}$

## Elementary function representations

$\pi = 4 \operatorname{atan}\!\left(1\right)$
$\pi = 16 \operatorname{acot}\!\left(5\right) - 4 \operatorname{acot}\!\left(239\right)$
$\pi = -i \log\!\left(-1\right)$

## Integral representations

$\pi = 2 \int_{-1}^{1} \sqrt{1 - {x}^{2}} \, dx$
$\pi = \int_{-\infty}^{\infty} \frac{1}{{x}^{2} + 1} \, dx$
$\pi = {\left(\int_{-\infty}^{\infty} {e}^{-{x}^{2}} \, dx\right)}^{2}$

## Series representations

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

## Product representations

$\pi = 2 \prod_{k=1}^{\infty} \frac{4 {k}^{2}}{4 {k}^{2} - 1}$

## Limit representations

$\pi = \lim_{k \to \infty} \frac{{16}^{k}}{k {{2 k \choose k}}^{2}}$

## Approximations

$\left|\pi - \frac{22}{7}\right| \lt 0.00127$
$\left|\pi - \frac{355}{113}\right| \lt 2.67 \cdot 10^{-7}$
$\left|\pi - \frac{\log\!\left({640320}^{3} + 744\right)}{\sqrt{163}}\right| \lt 2.24 \cdot 10^{-31}$
$\left|\frac{1}{\pi} - \left(12 \sum_{k=0}^{N - 1} \frac{{\left(-1\right)}^{k} \left(6 k\right)! \left(13591409 + 545140134 k\right)}{\left(3 k\right)! {\left(k !\right)}^{3} {640320}^{3 k + 3 / 2}}\right)\right| \lt \frac{1}{{151931373056000}^{N}}$

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

2019-05-23 08:00:13.607731 UTC