BarnesG

The Barnes G-function $G(z)$, the double-gamma function satisfying $G(z+1) = \Gamma(z)\,G(z)$ with $G(1) = 1$ — so $G(n) = \prod_{k=0}^{n-2} k!$, the superfactorial, at positive integers.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaBarnes G-function↗MathWorldBarnesG-Function↗WikidataQ808463↗DLMF5.17.E1↗FungrimBarnesG✓↗Fungrim entryWolfram LanguageBarnesG!↗

BarnesG(z)the Barnes G-function $G(z)$.

mpmathbarnesg(z)Juliaenumeratio_ball(barnes_g, z)

Typed signatures

Type	Package	Applies
(complex | infinity) -> number	analyticreplaces compute-engine	always
(complex | infinity) -> number	compute-engine	always

Domain: Special functions

Details

The Barnes G-function is the double gamma function: it satisfies $G(z+1) = \Gamma(z)\,G(z)$ with $G(1) = 1$ (DLMF 5.17.1), one level up from $\Gamma(z+1) = z\,\Gamma(z)$, and carries the Weierstrass-type product $G(z+1) = (2\pi)^{z/2} \exp\!\left(-\tfrac12 z(z+1) - \tfrac12\gamma z^2\right) \prod_{k=1}^\infty \left(1+\tfrac{z}{k}\right)^k \exp\!\left(-z+\tfrac{z^2}{2k}\right)$ (DLMF 5.17.3) that entire functions are built from.

• Functional equation $G(z+1) = \Gamma(z)\,G(z)$, the analogue of $\Gamma(z+1) = z\,\Gamma(z)$ one level up; see Gamma.
• At positive integers $G(n) = 0!\,1!\,2!\cdots(n-2)!$ — the superfactorial: $G(1) = G(2) = G(3) = 1$, $G(4) = 2$, $G(5) = 12$, $G(6) = 288$, $G(7) = 34560$. Exact and arbitrarily large.
• Zeros at the nonpositive integers $0, -1, -2, \dots$ (where $\Gamma$ has poles); entire, with no poles of its own.
• Numeric evaluation exponentiates LogBarnesG: the asymptotic series for $\ln G(z+1)$ for large $\operatorname{Re}(z)$, reached through the functional equation in logarithms. Complex $z$ supported; aligned with Wolfram's $\mathrm{BarnesG}[z]$.
• The value itself overflows a double past $|z| \approx 60$ (G(60) $\approx 4.6\times10^{1971}$); use LogBarnesG there, or stay exact at integers.

Plot

ComplexPlot

real slice

Examples

11 more kept as data, checked against the oracles

Basic #

$G(5) = 0!\,1!\,2!\,3! = 12$

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In$\mathrm{BarnesG}(5)$Out$12$

$G(7) = 34560$

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In$\mathrm{BarnesG}(7)$Out$34\,560$

The superfactorials, threaded over a list

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In$\mathrm{BarnesG}(\bigl\lbrack1, 2, 3, 4, 5, 6\bigr\rbrack)$Out$\bigl\lbrack1, 1, 1, 2, 12, 288\bigr\rbrack$

A floating-point argument evaluates numerically

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In$\mathrm{BarnesG}(2.5)$Out$0.947\,573\,901\,084\,062\,7$

$G(10) = \prod_{k=0}^{8} k!$, exact

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In$\mathrm{BarnesG}(10)$Out$5\,056\,584\,744\,960\,000$

$G(3) = 0!\,1! = 1$

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In$\mathrm{BarnesG}(3)$Out$1$

$G(\tfrac12)$ to machine precision

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In$\mathrm{BarnesG}(0.5)$Out$0.603\,244\,281\,209\,589\,3$

Scope #

Complex arguments

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In$\mathrm{BarnesG}(1.3+i)$Out$1.444\,613\,935\,050\,315\,3-0.298\,548\,825\,211\,568\,7i$

Negative non-integer arguments, between the zeros

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In$\mathrm{BarnesG}(-2.5)$Out$0.076\,172\,979\,656\,884\,05$

Threads elementwise over an array

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In$\mathrm{BarnesG}(\bigl\lbrack\bigl\lbrack\frac{1}{2}, 1\bigr\rbrack, \bigl\lbrack1, \frac{1}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\mathrm{BarnesG}(\frac{1}{2}), 1\bigr\rbrack, \bigl\lbrack1, \mathrm{BarnesG}(\frac{1}{2})\bigr\rbrack\bigr\rbrack$

$G(\infty) = \infty$

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In$\mathrm{BarnesG}(∞)$Out$∞$

Interval arithmetic, proven: $G$'s certified derivative keeps one sign on $[1.17, 1.18]$, so the image is its values at the ends, rounded outward

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In$\mathrm{BarnesG}(\mathrm{Interval}(1.17, 1.18))$Out$\mathrm{Interval}(1.052\,216\,328\,686\,19, 1.054\,193\,606\,435\,7)$

Uncertainty propagation: $G(2.1 \pm 0.01) = 0.98477 \pm 0.00145$, to first order in $G'$

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In$\mathrm{BarnesG}(\mathrm{Around}(2.1, 0.01))$Out$\mathrm{Around}(0.984\,772\,724\,354\,046\,5, 0.001\,449\,505\,695\,943\,639)$

To 30 significant digits, past a double's reach

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In$N(\mathrm{BarnesG}(\frac{1}{2}), 30)$Out$0.603\,244\,281\,209\,446\,206\,191\,429\,224\,535$

Properties #

$G$ vanishes at the nonpositive integers

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In$\mathrm{BarnesG}(0)$Out$0$

…all of them

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In$\mathrm{BarnesG}(-3)$Out$0$

The hyperfactorial $H(n) = \Gamma(n+1)^n / G(n+1)$: here $120^5/288 = H(5)$

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In$\frac{Γ(6)^5}{\mathrm{BarnesG}(6)}$Out$86\,400\,000$

Possible issues #

An exact non-integer argument stays symbolic under plain evaluation (use N(); Wolfram's closed form in Glaisher's constant is not reproduced)

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In$\mathrm{BarnesG}(\frac{5}{2})$Out$\mathrm{BarnesG}(\frac{5}{2})$

About

Examples	29
Checked against	Wolfram Language 20 agree9 differmpmath 21 agreeJulia 20 agree1 error
Crosswalk	WikipediaMathWorldWikidataDLMFFungrimFungrim entry ×12Wolfram LanguagempmathJulia

Implementation

referenceenginenotatio

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

nativeenginetypescriptpackages/ce-patches/src/compute-engine/numerics/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.

mappedmpmathpackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

See also: LogBarnesG, Gamma, LogGamma, Factorial