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Catalan's constant

Table of contents: Numerical value - Special function representations - Series representations - Integral representations

f7b6aa
Symbol: ConstCatalan GG Catalan's constant

Numerical value

6a83ad
G[0.91596559417721901505460351493238411077414937428167±2.141051]G \in \left[0.91596559417721901505460351493238411077414937428167 \pm 2.14 \cdot 10^{-51}\right]

Special function representations

2744d4
G=18(ψ ⁣(14)π2)G = \frac{1}{8} \left(\psi'\!\left(\frac{1}{4}\right) - {\pi}^{2}\right)
e85723
G=116(ζ ⁣(2,14)ζ ⁣(2,34))G = \frac{1}{16} \left(\zeta\!\left(2, \frac{1}{4}\right) - \zeta\!\left(2, \frac{3}{4}\right)\right)
1d65c2
G=Im ⁣(Li2 ⁣(i))G = \operatorname{Im}\!\left(\operatorname{Li}_{2}\!\left(i\right)\right)
9e9922
G=L ⁣(2,χ4.3)G = L\!\left(2, \chi_{4 \, . \, 3}\right)
4c166d
G=14Φ ⁣(1,2,12)G = \frac{1}{4} \Phi\!\left(-1, 2, \frac{1}{2}\right)
a766f2
G=3F2 ⁣(12,12,1,32,32,1)G = \,{}_3F_2\!\left(\frac{1}{2}, \frac{1}{2}, 1, \frac{3}{2}, \frac{3}{2}, -1\right)

Series representations

33aa62
G=n=0(1)n(2n+1)2G = \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{{\left(2 n + 1\right)}^{2}}
d43f30
G=12n=04n(2n+1)2(2nn)G = \frac{1}{2} \sum_{n=0}^{\infty} \frac{{4}^{n}}{{\left(2 n + 1\right)}^{2} {2 n \choose n}}
0bd544
G=π8log ⁣(2+3)+38n=01(2n+1)2(2nn)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} {2 n \choose n}}
37fb5f
G=164n=1256n(580n2184n+15)n3(2n1)(6n3n)(6n4n)(4n2n)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}}
a8657e
G=1n=1nζ ⁣(2n+1)16nG = 1 - \sum_{n=1}^{\infty} \frac{n \zeta\!\left(2 n + 1\right)}{{16}^{n}}

Integral representations

Involving elementary functions

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G=01atan(x)xdxG = \int_{0}^{1} \frac{\operatorname{atan}(x)}{x} \, dx
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G=01log(x)x2+1dxG = -\int_{0}^{1} \frac{\log(x)}{{x}^{2} + 1} \, dx
49df16
G=1log(x)x2+1dxG = \int_{1}^{\infty} \frac{\log(x)}{{x}^{2} + 1} \, dx
997777
G=π216+πlog(2)401(atan(x))2dxG = \frac{{\pi}^{2}}{16} + \frac{\pi \log(2)}{4} - \int_{0}^{1} {\left(\operatorname{atan}(x)\right)}^{2} \, dx
d6703a
G=7ζ(3)4π+2π01(atan(x))2xdxG = \frac{7 \zeta(3)}{4 \pi} + \frac{2}{\pi} \int_{0}^{1} \frac{{\left(\operatorname{atan}(x)\right)}^{2}}{x} \, dx
fd82ab
G=01acos(x)x2+1dxG = \int_{0}^{1} \frac{\operatorname{acos}(x)}{\sqrt{{x}^{2} + 1}} \, dx
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G=01asinh(x)1x2dxG = \int_{0}^{1} \frac{\operatorname{asinh}(x)}{\sqrt{1 - {x}^{2}}} \, dx
c54c85
G=120xcosh(x)dxG = \frac{1}{2} \int_{0}^{\infty} \frac{x}{\cosh(x)} \, dx
ec1435
G=14π/2π/2xsin(x)dxG = \frac{1}{4} \int_{-\pi / 2}^{\pi / 2} \frac{x}{\sin(x)} \, dx
79f20e
G=0π/4xsin(x)cos(x)dxG = \int_{0}^{\pi / 4} \frac{x}{\sin(x) \cos(x)} \, dx

Involving compositions of elementary functions

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G=0atan ⁣(ex)dxG = \int_{0}^{\infty} \operatorname{atan}\!\left({e}^{-x}\right) \, dx
08cda4
G=0π/4log ⁣(tan(x))dxG = -\int_{0}^{\pi / 4} \log\!\left(\tan(x)\right) \, dx
270e67
G=0π/4log ⁣(cot(x))dxG = \int_{0}^{\pi / 4} \log\!\left(\cot(x)\right) \, dx
4dec89
G=0π/2asinh ⁣(sin(x))dxG = \int_{0}^{\pi / 2} \operatorname{asinh}\!\left(\sin(x)\right) \, dx
d6415e
G=0π/2asinh ⁣(cos(x))dxG = \int_{0}^{\pi / 2} \operatorname{asinh}\!\left(\cos(x)\right) \, dx
e09b77
G=20π/4log ⁣(2sin(x))dxG = -2 \int_{0}^{\pi / 4} \log\!\left(2 \sin(x)\right) \, dx
6d3591
G=20π/4log ⁣(2cos(x))dxG = 2 \int_{0}^{\pi / 4} \log\!\left(2 \cos(x)\right) \, dx

Involving special functions

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G=1201K ⁣(m2)dmG = \frac{1}{2} \int_{0}^{1} K\!\left({m}^{2}\right) \, dm
d3cfc2
G=01E ⁣(m2)dm12G = \int_{0}^{1} E\!\left({m}^{2}\right) \, dm - \frac{1}{2}
937fa9
G=π201/2Γ ⁣(1+x)Γ ⁣(1x)dxG = \frac{\pi}{2} \int_{0}^{1 / 2} \Gamma\!\left(1 + x\right) \Gamma\!\left(1 - x\right) \, dx

Double integrals

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G=010111+x2y2dxdyG = \int_{0}^{1} \int_{0}^{1} \frac{1}{1 + {x}^{2} {y}^{2}} \, dx \, dy
ed4cca
G=1401011(x+y)1x1ydxdyG = \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.

2019-10-05 13:11:19.856591 UTC