Definition:StieltjesConstants

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The LaTeX DLMF and DRMF macro \StieltjesConstants represents the Stiltjes constants Ξ³ n Stieltjes-constants 𝑛 {\displaystyle{\displaystyle\gamma_{n}}} .

This macro is in the category of constants.

In math mode, this macro can be called in the following way:

\StieltjesConstants{n} produces Ξ³ n Stieltjes-constants 𝑛 {\displaystyle{\displaystyle{\displaystyle\gamma_{n}}}}

These are defined by Ξ³ n := lim m β†’ ∞ ⁑ ( βˆ‘ k = 1 m ( ln ⁑ k ) n k - ( ln ⁑ m ) n + 1 n + 1 ) . assign Stieltjes-constants 𝑛 subscript β†’ π‘š superscript subscript π‘˜ 1 π‘š superscript π‘˜ 𝑛 π‘˜ superscript π‘š 𝑛 1 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\gamma_{n}:=\lim_{m\to\infty}\left(% \sum_{k=1}^{m}\frac{(\ln k)^{n}}{k}-\frac{(\ln m)^{n+1}}{n+1}\right).}}}

The Stieltjes constant Ξ³ 0 = 0.57721 56649 01532 86 ⁒ … Stieltjes-constants 0 0.57721 56649 01532 86 … {\displaystyle{\displaystyle\gamma_{0}=0.57721\;56649\;01532\;86...}} equals Ξ³ {\displaystyle{\displaystyle\gamma}} the Euler-Mascheroni constant.

Bibliography

Equation (2), Section 2.21 of FI.

http://en.wikipedia.org/wiki/Stieltjes_constants

Symbols List

Ξ³ n subscript 𝛾 𝑛 {\displaystyle{\displaystyle{\displaystyle\gamma_{n}}}}  : Stieltjes constants : http://drmf.wmflabs.org/wiki/Definition:StieltjesConstants
Ξ£ Ξ£ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
ln ln {\displaystyle{\displaystyle{\displaystyle\mathrm{ln}}}}  : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
Ξ³ 𝛾 {\displaystyle{\displaystyle{\displaystyle\gamma}}}  : Euler's (Euler-Mascheroni) constant : http://dlmf.nist.gov/5.2#E3


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