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Catalan's constant

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

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Symbol: ConstCatalan — GG — Catalan's constant

Numerical value

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G∈[0.91596559417721901505460351493238411077414937428167±2.14⋅10−51]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⁡ ⁣(Li⁡2 ⁣(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

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G=∑n=0∞(−1)n(2n+1)2G = \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{{\left(2 n + 1\right)}^{2}}
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G=12∑n=0∞4n(2n+1)2⋅(2nn)G = \frac{1}{2} \sum_{n=0}^{\infty} \frac{{4}^{n}}{{\left(2 n + 1\right)}^{2} \cdot {2 n \choose n}}
0bd544
G=π8log⁡ ⁣(2+3)+38∑n=0∞1(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} \cdot {2 n \choose n}}
37fb5f
G=164∑n=1∞256n(580n2−184n+15)n3(2n−1)(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=1−∑n=1∞nζ ⁣(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)x dxG = \int_{0}^{1} \frac{\operatorname{atan}(x)}{x} \, dx
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G=−∫01log⁡(x)x2+1 dxG = -\int_{0}^{1} \frac{\log(x)}{{x}^{2} + 1} \, dx
49df16
G=∫1∞log⁡(x)x2+1 dxG = \int_{1}^{\infty} \frac{\log(x)}{{x}^{2} + 1} \, dx
997777
G=π216+πlog⁡(2)4−∫01(atan⁡(x))2 dxG = \frac{{\pi}^{2}}{16} + \frac{\pi \log(2)}{4} - \int_{0}^{1} {\left(\operatorname{atan}(x)\right)}^{2} \, dx
d6703a
G=7ζ ⁣(3)4π+2π∫01(atan⁡(x))2x dxG = \frac{7 \zeta\!\left(3\right)}{4 \pi} + \frac{2}{\pi} \int_{0}^{1} \frac{{\left(\operatorname{atan}(x)\right)}^{2}}{x} \, dx
fd82ab
G=∫01acos⁡(x)x2+1 dxG = \int_{0}^{1} \frac{\operatorname{acos}(x)}{\sqrt{{x}^{2} + 1}} \, dx
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G=∫01asinh⁡(x)1−x2 dxG = \int_{0}^{1} \frac{\operatorname{asinh}(x)}{\sqrt{1 - {x}^{2}}} \, dx
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G=12∫0∞xcosh⁡(x) dxG = \frac{1}{2} \int_{0}^{\infty} \frac{x}{\cosh(x)} \, dx
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G=14∫−π/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=∫0∞atan⁡ ⁣(e−x) 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
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G=∫0π/2asinh⁡ ⁣(cos⁡(x)) dxG = \int_{0}^{\pi / 2} \operatorname{asinh}\!\left(\cos(x)\right) \, dx
e09b77
G=−2∫0π/4log⁡ ⁣(2sin⁡(x)) dxG = -2 \int_{0}^{\pi / 4} \log\!\left(2 \sin(x)\right) \, dx
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G=2∫0π/4log⁡ ⁣(2cos⁡(x)) dxG = 2 \int_{0}^{\pi / 4} \log\!\left(2 \cos(x)\right) \, dx

Involving special functions

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G=12∫01K ⁣(m2) dmG = \frac{1}{2} \int_{0}^{1} K\!\left({m}^{2}\right) \, dm
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G=∫01E ⁣(m2) dm−12G = \int_{0}^{1} E\!\left({m}^{2}\right) \, dm - \frac{1}{2}
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G=π2∫01/2Γ ⁣(1+x)Γ ⁣(1−x) 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=∫01∫0111+x2y2 dx dyG = \int_{0}^{1} \int_{0}^{1} \frac{1}{1 + {x}^{2} {y}^{2}} \, dx \, dy
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G=14∫01∫011(x+y)1−x1−y dx dyG = \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