enumeratio

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StieltjesGamma ​

The Stieltjes constants , the coefficients of the Laurent expansion of at : ; with a second argument, the generalized for HurwitzZeta. Provided by `@enumeratio/analytic`.

WikipediaStieltjes constantsMathWorldStieltjesConstantsWikidataQ1816316DLMF25.2.E5FungrimStieltjesGamma✓Fungrim entryWolfram LanguageStieltjesGamma
StieltjesGamma(n)the -th Stieltjes constant .
StieltjesGamma(n, a)the generalized Stieltjes constant , from the expansion of at .

Domain: Special functions

Details
  • is Euler's constant; , . They have no closed forms beyond .
  • ; for this is the definition of Euler's constant.
  • Generalized: , so and , the negated digamma. Shift: . Poles at .
  • Same normalisation as Wolfram's and mpmath's `stieltjes(n, a)`. Complex supported.
  • Numerically, Euler–Maclaurin on . In double precision the partial sum and the subtracted term cancel more and more with : about relative through , at , at — and orders past 30 are left unevaluated rather than returned wrong.

Examples

Implementation

primitive · numericirreducibly numeric — it evaluates rather than rewrites

nativeenginetypescriptpackages/analytic/src/stieltjes.ts
mappedexternalwolfram / mpmathpackages/oracle/src/mappings.ts

StieltjesGamma[n, a]; mpmath.stieltjes(n, a).

See also: Zeta, HurwitzZeta, EulerGamma, Digamma