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StieltjesGamma

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

StieltjesGamma[n]

gives the n-th Stieltjes constant gamma_n, the Laurent coefficients of

Notes Zeta about s = 1. StieltjesGamma\[0\] is EulerGamma; higher constants are inert (they stay symbolic) and appear in Series expansions of Zeta. Listable.

Examples (5)

Every input below was run against the current Mathilda build and its output recorded.

Basic examples (1)

In[1]:= StieltjesGamma[0]
Out[1]= EulerGamma

Applications (4)

In[2]:= StieltjesGamma[0]
Out[2]= EulerGamma

In[3]:= StieltjesGamma[3]
Out[3]= StieltjesGamma[3]

In[4]:= N[StieltjesGamma[0], 40]
Out[4]= 0.57721566490153286060651209008240243104214

In[5]:= Series[Zeta[s], {s, 1, 2}]
Out[5]= 1/(s - 1) + EulerGamma + -StieltjesGamma[1] (s - 1) + 1/2 StieltjesGamma[2] (s - 1)^2 + O[s - 1]^3

Algorithm

Mathilda -- StieltjesGamma, the Stieltjes constants gamma_n.

  StieltjesGamma[n] = gamma_n, the coefficients of the Laurent expansion
  of the Riemann zeta function about s = 1:

    zeta(s) = 1/(s-1) + Sum_{n>=0} ((-1)^n / n!) gamma_n (s-1)^n.

gamma_0 is the Euler-Mascheroni constant EulerGamma. The higher constants have no elementary closed form, so StieltjesGamma is inert: it stays symbolic, except for the single reduction StieltjesGamma[0] -> EulerGamma. It is the natural output of Series[Zeta[x], {x, 1, n}] (and the Taylor expansion of Zeta about 0).

Like LogGamma (see src/polygamma.c), this module owns only the symbol's identity and the n = 0 reduction; all generic symbolic behaviour comes from the evaluator.

Attributes: Listable, Protected.

Implementation notes

Attributes: Listable, Protected.

References

See also: Zeta, EulerGamma, Series

Notes & additional examples

Notes

StieltjesGamma[n] denotes the n-th Stieltjes constant γ_n, defined by the Laurent expansion Zeta[s] = 1/(s - 1) + Sum[(-1)^n/n! γ_n (s - 1)^n] about the pole at s = 1. StieltjesGamma[0] is EulerGamma; the higher constants are inert symbols that appear, with the correct (-1)^n/n! factors, in the Series expansion of Zeta at s = 1. It is Listable.