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Stieltjes constants

Table of contents: Definitions - Generating functions - Limit representations - Specific values - Recurrence relations - Integral representations - Bounds and inequalities

Definitions

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Symbol: StieltjesGamma γn ⁣(a)\gamma_{n}\!\left(a\right) Stieltjes constant

Generating functions

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ζ(s)=1s1+n=0(1)nn!γn(s1)n\zeta(s) = \frac{1}{s - 1} + \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{n !} \gamma_{n} {\left(s - 1\right)}^{n}
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ζ ⁣(s,a)=1s1+n=0(1)nn!γn ⁣(a)(s1)n\zeta\!\left(s, a\right) = \frac{1}{s - 1} + \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{n !} \gamma_{n}\!\left(a\right) {\left(s - 1\right)}^{n}

Limit representations

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γn ⁣(a)=limN[(k=0Nlogn ⁣(k+a)k+a)logn+1 ⁣(N+a)n+1]\gamma_{n}\!\left(a\right) = \lim_{N \to \infty} \left[\left(\sum_{k=0}^{N} \frac{\log^{n}\!\left(k + a\right)}{k + a}\right) - \frac{\log^{n + 1}\!\left(N + a\right)}{n + 1}\right]

Specific values

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γn ⁣(1)=γn\gamma_{n}\!\left(1\right) = \gamma_{n}
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γ0 ⁣(1)=γ0=γ\gamma_{0}\!\left(1\right) = \gamma_{0} = \gamma
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γ0 ⁣(a)=ψ ⁣(a)\gamma_{0}\!\left(a\right) = -\psi\!\left(a\right)
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γ1 ⁣(12)=γ12γlog(2)log2 ⁣(2)\gamma_{1}\!\left(\frac{1}{2}\right) = \gamma_{1} - 2 \gamma \log(2) - \log^{2}\!\left(2\right)
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Table of γn\gamma_{n} to 50 digits for 0n300 \le n \le 30
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Table of γ10n\gamma_{{10}^{n}} to 50 digits for 0n200 \le n \le 20

Recurrence relations

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γn ⁣(a+1)=γn ⁣(a)logn ⁣(a)a\gamma_{n}\!\left(a + 1\right) = \gamma_{n}\!\left(a\right) - \frac{\log^{n}\!\left(a\right)}{a}

Integral representations

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γn ⁣(a)=π2(n+1)0logn+1 ⁣(a12+ix)+logn+1 ⁣(a12ix)cosh2 ⁣(πx)dx\gamma_{n}\!\left(a\right) = -\frac{\pi}{2 \left(n + 1\right)} \int_{0}^{\infty} \frac{\log^{n + 1}\!\left(a - \frac{1}{2} + i x\right) + \log^{n + 1}\!\left(a - \frac{1}{2} - i x\right)}{\cosh^{2}\!\left(\pi x\right)} \, dx

Bounds and inequalities

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γn<104exp ⁣(nlog ⁣(log(n)))\left|\gamma_{n}\right| < {10}^{-4} \exp\!\left(n \log\!\left(\log(n)\right)\right)

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

2019-11-11 15:50:15.016492 UTC