StieltjesGamma

StieltjesGamma[n]

gives the Stieltjes constant .

StieltjesGamma[n,a]

gives the generalized Stieltjes constant .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • is the coefficient of in the Laurent expansion of about the point .
  • The are generalizations of Euler's constant; .
  • is the coefficient of in the Laurent expansion of about the point .
  • For certain special arguments, StieltjesGamma automatically evaluates to exact values.
  • StieltjesGamma can be evaluated to arbitrary numerical precision.
  • StieltjesGamma automatically threads over lists.

Examples

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Basic Examples  (3)

Evaluate to high precision:

Plot values of StieltjesGamma:

Scope  (4)

TraditionalForm formatting:

Evaluate for complex second argument:

The precision of the output tracks the precision of the input:

StieltjesGamma threads element-wise over lists:

Applications  (3)

Expansion of the Riemann zeta function:

Expansion of the Hurwitz zeta function:

Test Lis criterion for the Riemann hypothesis:

All values should be positive:

Express integrals in terms of StieltjesGamma:

Properties & Relations  (2)

The EulerGamma case evaluates automatically:

Various symbolic relations are automatically used:

Possible Issues  (4)

Substitution of derivatives of Zeta at yields indeterminate values:

Use Limit to obtain the expansion coefficient:

The argument of StieltjesGamma must be an exact non-negative integer:

Use N to obtain a numerical approximation:

Alternatively, use two-argument form:

StieltjesGamma does not allow numericalization of its index:

It is currently not known if Stieltjes constants are algebraic numbers:

Introduced in 1996
 (3.0)
 |
Updated in 2008
 (7.0)