# 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

$\gamma = \lim_{n \to \infty} \left[\left(\sum_{k=1}^{n} \frac{1}{k}\right) - \log(n)\right]$
${e}^{\gamma} = \lim_{N \to \infty} \frac{1}{\log\!\left(p_{N}\right)} \prod_{n=1}^{N} \frac{p_{n}}{p_{n} - 1}$

## Special function representations

### Direct values

$\gamma = -\Gamma'(1)$
$\gamma = -\psi\!\left(1\right)$

### Limits at singularities

$\gamma = \lim_{s \to 1} \left[\zeta\!\left(s\right) - \frac{1}{s - 1}\right]$
$\gamma = -\lim_{z \to 0} \left[\Gamma(z) - \frac{1}{z}\right]$
$\gamma = -\lim_{z \to 0} \left[\psi\!\left(z\right) + \frac{1}{z}\right]$
$\gamma = \lim_{x \to {0}^{+}} \left[\frac{\pi}{2} Y_{0}\!\left(x\right) - \log\!\left(\frac{x}{2}\right)\right]$
$\gamma = \lim_{x \to {0}^{+}} \left[-K_{0}\!\left(x\right) - \log\!\left(\frac{x}{2}\right)\right]$

### Infinite series

$\gamma = 1 - \sum_{k=2}^{\infty} \frac{\zeta\!\left(k\right) - 1}{k}$

## Integral representations

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

## Approximations

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

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.

2020-04-08 16:14:44.404316 UTC