StieltjesGamma

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.
• StieltjesGamma can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples(3)

Evaluate to high precision:

Plot values of StieltjesGamma:

Scope(5)

Evaluate for complex second argument:

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

StieltjesGamma can be used with Interval and CenteredInterval objects:

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:

Wolfram Research (1996), StieltjesGamma, Wolfram Language function, https://reference.wolfram.com/language/ref/StieltjesGamma.html (updated 2022).

Text

Wolfram Research (1996), StieltjesGamma, Wolfram Language function, https://reference.wolfram.com/language/ref/StieltjesGamma.html (updated 2022).

CMS

Wolfram Language. 1996. "StieltjesGamma." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/StieltjesGamma.html.

APA

Wolfram Language. (1996). StieltjesGamma. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/StieltjesGamma.html

BibTeX

@misc{reference.wolfram_2023_stieltjesgamma, author="Wolfram Research", title="{StieltjesGamma}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/StieltjesGamma.html}", note=[Accessed: 28-September-2023 ]}

BibLaTeX

@online{reference.wolfram_2023_stieltjesgamma, organization={Wolfram Research}, title={StieltjesGamma}, year={2022}, url={https://reference.wolfram.com/language/ref/StieltjesGamma.html}, note=[Accessed: 28-September-2023 ]}