# Euler's constant

Symbol: ConstGamma $\gamma$ The constant gamma (0.577...)

## Numerical value

$\gamma \in \left[0.57721566490153286060651209008240243104215933593992 \pm 3.60 \cdot 10^{-51}\right]$
$\gamma \notin \left\{ \frac{p}{q} : p \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, q \le {10}^{242080} \right\}$

## Limit representations

$\gamma = \lim_{n \to \infty} \sum_{k=1}^{n} \frac{1}{k} - \log\!\left(n\right)$

## Special function representations

$\gamma = \lim_{s \to 1} \zeta\!\left(s\right) - \frac{1}{s - 1}$
$\gamma = -\Gamma'(1)$
$\gamma = -\psi\!\left(1\right)$
$\gamma = 1 - \sum_{k=2}^{\infty} \frac{\zeta\!\left(k\right) - 1}{k}$

## Integral representations

$\gamma = -\int_{0}^{\infty} {e}^{-x} \log\!\left(x\right) \, dx$
$\gamma = -\int_{0}^{1} \log\!\left(\log\!\left(\frac{1}{x}\right)\right) \, dx$

## Approximations

$\left|\gamma - \left(\frac{S}{I} - \frac{T}{{I}^{2}} - \log\!\left(n\right)\right)\right| \lt 24 {e}^{-8 n}\; \text{ where } \left(S, I, T\right) = \left(\sum_{k=0}^{5 n} \frac{H_{k} {n}^{2 k}}{{\left(k !\right)}^{2}}, \sum_{k=0}^{5 n} \frac{{n}^{2 k}}{{\left(k !\right)}^{2}}, \frac{1}{4 n} \sum_{k=0}^{2 n - 1} \frac{{\left(\left(2 k\right)!\right)}^{3}}{{\left(k !\right)}^{4} {8}^{2 k} {\left(2 n\right)}^{2 k}}\right)$

Related topics: Gamma function, Riemann zeta function

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

2019-06-18 07:49:59.356594 UTC