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# Catalan's constant

Symbol: ConstCatalan $G$ Catalan's constant

## Numerical value

$G \in \left[0.91596559417721901505460351493238411077414937428167 \pm 2.14 \cdot 10^{-51}\right]$

## Special function representations

$G = \frac{1}{8} \left(\psi'\!\left(\frac{1}{4}\right) - {\pi}^{2}\right)$
$G = \frac{1}{16} \left(\zeta\!\left(2, \frac{1}{4}\right) - \zeta\!\left(2, \frac{3}{4}\right)\right)$
$G = \operatorname{Im}\!\left(\operatorname{Li}_{2}\!\left(i\right)\right)$
$G = L\!\left(2, \chi_{4 \, . \, 3}\right)$
$G = \frac{1}{4} \Phi\!\left(-1, 2, \frac{1}{2}\right)$
$G = \,{}_3F_2\!\left(\frac{1}{2}, \frac{1}{2}, 1, \frac{3}{2}, \frac{3}{2}, -1\right)$

## Series representations

$G = \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{{\left(2 n + 1\right)}^{2}}$
$G = \frac{1}{2} \sum_{n=0}^{\infty} \frac{{4}^{n}}{{\left(2 n + 1\right)}^{2} \cdot {2 n \choose n}}$
$G = \frac{\pi}{8} \log\!\left(2 + \sqrt{3}\right) + \frac{3}{8} \sum_{n=0}^{\infty} \frac{1}{{\left(2 n + 1\right)}^{2} \cdot {2 n \choose n}}$
$G = \frac{1}{64} \sum_{n=1}^{\infty} \frac{{256}^{n} \left(580 {n}^{2} - 184 n + 15\right)}{{n}^{3} \left(2 n - 1\right) {6 n \choose 3 n} {6 n \choose 4 n} {4 n \choose 2 n}}$
$G = 1 - \sum_{n=1}^{\infty} \frac{n \zeta\!\left(2 n + 1\right)}{{16}^{n}}$

## Integral representations

### Involving elementary functions

$G = \int_{0}^{1} \frac{\operatorname{atan}(x)}{x} \, dx$
$G = -\int_{0}^{1} \frac{\log(x)}{{x}^{2} + 1} \, dx$
$G = \int_{1}^{\infty} \frac{\log(x)}{{x}^{2} + 1} \, dx$
$G = \frac{{\pi}^{2}}{16} + \frac{\pi \log(2)}{4} - \int_{0}^{1} {\left(\operatorname{atan}(x)\right)}^{2} \, dx$
$G = \frac{7 \zeta\!\left(3\right)}{4 \pi} + \frac{2}{\pi} \int_{0}^{1} \frac{{\left(\operatorname{atan}(x)\right)}^{2}}{x} \, dx$
$G = \int_{0}^{1} \frac{\operatorname{acos}(x)}{\sqrt{{x}^{2} + 1}} \, dx$
$G = \int_{0}^{1} \frac{\operatorname{asinh}(x)}{\sqrt{1 - {x}^{2}}} \, dx$
$G = \frac{1}{2} \int_{0}^{\infty} \frac{x}{\cosh(x)} \, dx$
$G = \frac{1}{4} \int_{-\pi / 2}^{\pi / 2} \frac{x}{\sin(x)} \, dx$
$G = \int_{0}^{\pi / 4} \frac{x}{\sin(x) \cos(x)} \, dx$

### Involving compositions of elementary functions

$G = \int_{0}^{\infty} \operatorname{atan}\!\left({e}^{-x}\right) \, dx$
$G = -\int_{0}^{\pi / 4} \log\!\left(\tan(x)\right) \, dx$
$G = \int_{0}^{\pi / 4} \log\!\left(\cot(x)\right) \, dx$
$G = \int_{0}^{\pi / 2} \operatorname{asinh}\!\left(\sin(x)\right) \, dx$
$G = \int_{0}^{\pi / 2} \operatorname{asinh}\!\left(\cos(x)\right) \, dx$
$G = -2 \int_{0}^{\pi / 4} \log\!\left(2 \sin(x)\right) \, dx$
$G = 2 \int_{0}^{\pi / 4} \log\!\left(2 \cos(x)\right) \, dx$

### Involving special functions

$G = \frac{1}{2} \int_{0}^{1} K\!\left({m}^{2}\right) \, dm$
$G = \int_{0}^{1} E\!\left({m}^{2}\right) \, dm - \frac{1}{2}$
$G = \frac{\pi}{2} \int_{0}^{1 / 2} \Gamma\!\left(1 + x\right) \Gamma\!\left(1 - x\right) \, dx$

### Double integrals

$G = \int_{0}^{1} \int_{0}^{1} \frac{1}{1 + {x}^{2} {y}^{2}} \, dx \, dy$
$G = \frac{1}{4} \int_{0}^{1} \int_{0}^{1} \frac{1}{\left(x + y\right) \sqrt{1 - x} \sqrt{1 - y}} \, dx \, dy$

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

2021-03-15 19:12:00.328586 UTC