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BarnesG ​

The Barnes G-function , the double-gamma function satisfying with — so , the superfactorial, at positive integers. Provided by `@enumeratio/analytic`.

WikipediaBarnes G-functionMathWorldBarnesG-FunctionWikidataQ808463DLMF5.17.E1FungrimBarnesG✓Fungrim entryWolfram LanguageBarnesG
BarnesG(z)the Barnes G-function .

Domain: Special functions

Details
  • Functional equation , the analogue of one level up; see Gamma.
  • At positive integers — the superfactorial: , , , , . Exact and arbitrarily large.
  • Zeros at the nonpositive integers (where has poles); entire, with no poles of its own.
  • Numeric evaluation exponentiates LogBarnesG: the asymptotic series for for large , reached through the functional equation in logarithms. Complex supported; aligned with Wolfram's .
  • The value itself overflows a double past (G(60) ); use LogBarnesG there, or stay exact at integers.

Examples

Implementation

referenceenginenotatio

G = exp(ln G) over the Weierstrass series of LogBarnesG; see there for what it costs to evaluate.

nativeenginetypescriptpackages/analytic/src/barnes-g.ts

exp of the LogBarnesG kernel; exact superfactorials at the integers come from the head.

mappedexternalwolfram / mpmathpackages/oracle/src/mappings.ts

BarnesG[z] and mpmath.barnesg(z), same normalisation.

See also: LogBarnesG, Gamma, LogGamma, Factorial