LogBarnesG

The logarithm of the Barnes G-function, $\ln G(z)$, as an analytic continuation — the form that stays finite where $G$ itself overflows.

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

LogBarnesG(z)the log-Barnes function $\ln G(z)$, analytically continued.

Juliaenumeratio_ball(log_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

• $\ln G(z+1) \sim \left(\tfrac{z^2}{2} - \tfrac{1}{12}\right)\ln z - \tfrac{3z^2}{4} + \tfrac{z}{2}\ln 2\pi + \zeta'(-1) + \sum_{k\ge1} \dfrac{B_{2k+2}}{4k(k+1)\,z^{2k}}$ as $\operatorname{Re}(z) \to \infty$, with $\zeta'(-1) = \tfrac{1}{12} - \ln A$ ($A$ Glaisher's constant). This is the numeric kernel, reached through $\ln G(z) = \ln G(z+n) - \sum_{k<n} \ln\Gamma(z+k)$.
• It is the continuation, not $\ln$ of the value: on the negative real axis its imaginary part is a multiple of $\pi$ fixed by continuity from above, so $\mathrm{LogBarnesG}(-2.5) = -2.5747\ldots + 6\pi i$ while $\ln G(-2.5)$ taken literally would be real. This matches Wolfram's $\mathrm{LogBarnesG}$.
• At positive integers it reduces through the exact superfactorial: $\ln G(4) = \ln 2$, $\ln G(3) = 0$.
• $-\infty$ at the nonpositive integers, where $G$ vanishes.

Plot

ComplexPlot

real slice

Examples

2 more kept as data, checked against the oracles

Basic #

$\ln G(4) = \ln 2$

live ↗

In$\mathrm{LogBarnesG}(4)$Out$\ln(2)$

$\ln G(3) = \ln 1 = 0$

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In$\mathrm{LogBarnesG}(3)$Out$0$

Numeric for a floating-point argument

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In$\mathrm{LogBarnesG}(10.5)$Out$42.278\,883\,636\,795\,2$

$\ln G(\tfrac12)$

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In$\mathrm{LogBarnesG}(0.5)$Out$-0.505\,433\,054\,489\,458\,3$

$\ln G(0.7)$ — negative, since $G < 1$ on $(0, 1)$

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In$\mathrm{LogBarnesG}(0.7)$Out$-0.214\,589\,897\,075\,086\,36$

Scope #

Complex arguments

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In$\mathrm{LogBarnesG}(1.7+i)$Out$0.121\,378\,662\,628\,899\,29-0.313\,561\,761\,336\,856\,65i$

Threads elementwise over an array: $-\infty$ at the zeros of $G$

live ↗

In$\mathrm{LogBarnesG}(\bigl\lbrack\bigl\lbrack4, -1\bigr\rbrack, \bigl\lbrack0, 3\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\ln(2), -∞\bigr\rbrack, \bigl\lbrack-∞, 0\bigr\rbrack\bigr\rbrack$

Huge arguments, straight from the asymptotic series: $\ln G(10^{1000}) \approx 1.1505\times10^{2003}$

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In$N(\mathrm{LogBarnesG}(10^{1\,000}))$Out$1.150\,542\,546\,497\,022\,842\,01\cdot10^{2003}$

Interval arithmetic over an increasing stretch, proven by the certified derivative

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In$\mathrm{LogBarnesG}(\mathrm{Interval}(0.111, 0.112))$Out$\mathrm{Interval}(-2.106\,284\,277\,112\,4, -2.096\,646\,198\,881\,46)$

Uncertainty propagation through $\ln G$

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In$\mathrm{LogBarnesG}(\mathrm{Around}(1.2, 0.01))$Out$\mathrm{Around}(0.056\,208\,855\,599\,105\,68, 0.001\,611\,305\,538\,862\,350\,7)$

$\ln G(\infty) = \infty$

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In$\mathrm{LogBarnesG}(∞)$Out$∞$

Properties #

$G(0) = 0$, so its logarithm is $-\infty$

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In$\mathrm{LogBarnesG}(0)$Out$-∞$

Exact at the integers: $\ln G(10) = \ln 5056584744960000$, split into logarithms of its factors

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In$\mathrm{LogBarnesG}(10)$Out$2\ln(14)+3\ln(10)+9\ln(3)+18\ln(2)+\ln(5)$

Possible issues #

The continuation, not $\ln$ of the value: $G(-2.5) > 0$, yet the imaginary part is $6\pi$

live ↗

In$\mathrm{LogBarnesG}(-2.5)$Out$-2.574\,748\,476\,853\,131+18.849\,555\,921\,538\,76i$

About

Examples	16
Checked against	Wolfram Language 12 agree4 differJulia 9 agree3 error
Crosswalk	WikipediaMathWorldWikidataDLMFFungrimFungrim entry ×18Wolfram LanguageJulia

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/ce-patches/src/compute-engine/numerics/barnes-g.ts

mappedjuliapackages/oracle/src/mappings.ts

See also: BarnesG, LogGamma, Gamma