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LogBarnesG ​

The logarithm of the Barnes G-function, , as an analytic continuation — the form that stays finite where itself overflows. Provided by `@enumeratio/analytic`.

WikipediaBarnes G-functionMathWorldBarnesG-FunctionWikidataQ808463DLMF5.17FungrimLogBarnesG✓Fungrim entryWolfram LanguageLogBarnesG
LogBarnesG(z)the log-Barnes function , analytically continued.

Domain: Special functions

Details
  • as , with ( Glaisher's constant). This is the numeric kernel, reached through .
  • It is the continuation, not of the value: on the negative real axis its imaginary part is a multiple of fixed by continuity from above, so while taken literally would be real. This matches Wolfram's .
  • At positive integers it reduces through the exact superfactorial: , .
  • at the nonpositive integers, where vanishes.

Examples

Implementation

referenceenginenotatio

The Weierstrass product in logarithms. Its terms are O(w³/k²), so it converges — slowly: a few hundred terms for a dozen digits, against the kernel's asymptotic series.

nativeenginetypescriptpackages/analytic/src/barnes-g.ts

See also: BarnesG, LogGamma, Gamma