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Euler's constant

Table of contents: Numerical value - Limit representations - Special function representations - Integral representations - Approximations - Related topics

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Symbol: ConstGamma — γ\gamma — The constant gamma (0.577...)

Numerical value

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γ∈[0.57721566490153286060651209008240243104215933593992±3.60⋅10−51]\gamma \in \left[0.57721566490153286060651209008240243104215933593992 \pm 3.60 \cdot 10^{-51}\right]
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γ∉{pq:p∈Z  and⁡  q∈Z≥1  and⁡  q≤10242080}\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

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γ=lim⁡n→∞[(∑k=1n1k)−log⁡(n)]\gamma = \lim_{n \to \infty} \left[\left(\sum_{k=1}^{n} \frac{1}{k}\right) - \log(n)\right]
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eγ=lim⁡N→∞1log⁡ ⁣(pN)∏n=1Npnpn−1{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

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γ=−Γ′(1)\gamma = -\Gamma'(1)
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γ=−ψ ⁣(1)\gamma = -\psi\!\left(1\right)

Limits at singularities

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γ=lim⁡s→1[ζ ⁣(s)−1s−1]\gamma = \lim_{s \to 1} \left[\zeta\!\left(s\right) - \frac{1}{s - 1}\right]
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γ=−lim⁡z→0[Γ(z)−1z]\gamma = -\lim_{z \to 0} \left[\Gamma(z) - \frac{1}{z}\right]
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γ=−lim⁡z→0[ψ ⁣(z)+1z]\gamma = -\lim_{z \to 0} \left[\psi\!\left(z\right) + \frac{1}{z}\right]
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γ=lim⁡x→0+[π2Y0 ⁣(x)−log⁡ ⁣(x2)]\gamma = \lim_{x \to {0}^{+}} \left[\frac{\pi}{2} Y_{0}\!\left(x\right) - \log\!\left(\frac{x}{2}\right)\right]
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γ=lim⁡x→0+[−K0 ⁣(x)−log⁡ ⁣(x2)]\gamma = \lim_{x \to {0}^{+}} \left[-K_{0}\!\left(x\right) - \log\!\left(\frac{x}{2}\right)\right]

Infinite series

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γ=1−∑k=2∞ζ ⁣(k)−1k\gamma = 1 - \sum_{k=2}^{\infty} \frac{\zeta\!\left(k\right) - 1}{k}

Integral representations

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γ=−∫0∞e−xlog⁡(x) dx\gamma = -\int_{0}^{\infty} {e}^{-x} \log(x) \, dx
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γ=−∫01log⁡ ⁣(log⁡ ⁣(1x)) dx\gamma = -\int_{0}^{1} \log\!\left(\log\!\left(\frac{1}{x}\right)\right) \, dx

Approximations

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∣γ−(SI−TI2−log⁡(n))∣<24e−8n   where (S,I,T)=(∑k=05nHkn2k(k!)2,∑k=05nn2k(k!)2,14n∑k=02n−1((2k)!)3(k!)4⋅82k(2n)2k)\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

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2021-03-15 19:12:00.328586 UTC