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

Examples¶

All examples below are verified against the current Mathilda build.

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

Implementation notes¶

Attributes: Listable, Protected.

Implementation status¶

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

References¶

Source: src/info.c
Specification: docs/spec/builtins/special-functions.md

Notes & additional examples¶

Worked examples¶

The zeroth Stieltjes constant is the Euler–Mascheroni constant:

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

Higher constants are inert and stay symbolic:

In[1]:= StieltjesGamma[3]
Out[1]= StieltjesGamma[3]

Numericalizing the zeroth constant recovers γ to 40 digits:

In[1]:= N[StieltjesGamma[0], 40]
Out[1]= 0.57721566490153286060651209008240243104214

The constants are exactly the Laurent coefficients of Zeta about s = 1 — expanding the series exhibits them in their defining role:

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

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.