LogGamma

The log-gamma function $\ln\Gamma(z)$ as an analytic continuation (branch cut on $(-\infty, 0]$), which differs from $\ln(\Gamma(z))$ by multiples of $2\pi i$ off the positive axis.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaGamma function↗MathWorldLogGammaFunction↗GammaFunction↗WikidataQ190573↗Encyclopedia of MathematicsGamma-function↗nLabGamma function↗Britannicascience/gamma-function↗DLMF5.2.E1↗FungrimLogGamma↗Wolfram LanguageLogGamma✓↗

LogGamma(z)the log-gamma function $\ln\Gamma(z)$, analytically continued.

SageMathlog_gamma(z)SymPyloggamma(z)mpmathloggamma(z)Juliaenumeratio_ball(lgamma, z)Rustln_gamma(z)

Typed signatures

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

Domain: Special functions

Details

• For real $z > 0$ it is simply $\ln\Gamma(z)$: $\ln\Gamma(n) = \ln (n-1)!$, $\ln\Gamma(\tfrac12) = \tfrac12\ln\pi$ -- where it agrees with compute-engine's own GammaLn.
• Elsewhere it is the continuation across the upper and lower half-planes, continuous off $(-\infty, 0]$ — the convention Wolfram's $\mathrm{LogGamma}$ and mpmath's loggamma share. $\ln(\Gamma(z))$ with a principal logarithm jumps by $2\pi i$ wherever $\Gamma$ crosses the negative axis; this does not. GammaLn is that principal version, so the two genuinely differ: at $z = -5/2$, GammaLn is $-0.0562$ while LogGamma is $-0.0562 - 3\pi i$.
• $+\infty$ at the poles of $\Gamma$, the nonpositive integers.
• Numerically, Stirling's series for large $\operatorname{Re}(z)$ with the recurrence $\ln\Gamma(z) = \ln\Gamma(z+n) - \sum_{k<n}\ln(z+k)$ — in principal logarithms, which for $\operatorname{Im}(z) \ne 0$ is exactly the continuation. compute-engine has a complex Gamma but no LogGamma of its own.

Plot

ComplexPlot

real slice

Examples

4 more kept as data, checked against the oracles

Basic #

$\ln\Gamma(3) = \ln 2! = \ln 2$

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In$\mathrm{LogGamma}(3)$Out$\ln(2)$

$\ln\Gamma(1) = 0$

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In$\mathrm{LogGamma}(1)$Out$0$

$\ln\Gamma(2.5) = \ln 1.3293\ldots$ — to the engine's working precision, not a double: right of the imaginary axis this goes through compute-engine's own GammaLn

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In$\mathrm{LogGamma}(2.5)$Out$0.284\,682\,870\,472\,919\,159\,632$

$\ln\Gamma(5) = \ln 4! = \ln 24$, split into prime logarithms

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In$\mathrm{LogGamma}(5)$Out$3\ln(2)+\ln(3)$

Scope #

To the engine's working precision

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In$\mathrm{LogGamma}(2.2)$Out$0.096\,947\,466\,790\,638\,776\,492$

Complex arguments

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In$\mathrm{LogGamma}(2.5+3i)$Out$-1.470\,954\,610\,348\,842\,5+2.822\,615\,638\,260\,8i$

Threads elementwise over an array

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In$\mathrm{LogGamma}(\bigl\lbrack\bigl\lbrack\frac{1}{2}, -1\bigr\rbrack, \bigl\lbrack0, \frac{1}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{\ln(π)}{2}, ∞\bigr\rbrack, \bigl\lbrack∞, \frac{\ln(π)}{2}\bigr\rbrack\bigr\rbrack$

Exact negative half-integers: $\ln\Gamma(-\tfrac32) = \ln\tfrac{4\sqrt\pi}{3} - 2\pi i$

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In$\mathrm{LogGamma}(-\frac{3}{2})$Out$\ln(\frac{4\sqrt{π}}{3})-2i π$

$\ln\Gamma(\infty) = \infty$

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In$\mathrm{LogGamma}(∞)$Out$∞$

Huge arguments: $\ln\Gamma(10^{300}) \approx 6.898\times10^{302}$, from Stirling's series directly rather than through $\Gamma(10^{300})$ itself, which overflows a double long before its log would; the answer carries the engine's precision

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In$N(\mathrm{LogGamma}(10^{300}))$Out$6.897\,755\,278\,982\,137\,052\,05\cdot10^{302}$

Interval arithmetic: decreasing here, so the endpoints swap

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In$\mathrm{LogGamma}(\mathrm{Interval}(0.41, 0.42))$Out$\mathrm{Interval}(0.746\,863\,727\,128\,013\,3, 0.771\,422\,463\,394\,747\,7)$

Uncertainty propagation, with $\psi(1.2)$ as the slope

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In$\mathrm{LogGamma}(\mathrm{Around}(1.2, 0.01))$Out$\mathrm{Around}(-0.085\,374\,090\,003\,315\,85, 0.002\,890\,398\,965\,921\,883)$

To 30 significant digits, the last one correctly rounded

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In$N(\mathrm{LogGamma}(\frac{1}{3}), 30)$Out$0.985\,420\,646\,927\,767\,069\,187\,174\,036\,978$

$\ln|\Gamma(-5/2)|$ is correctly rounded, not carried through a double $\ln\sin\pi z$ reflection

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In$N(\mathrm{LogGamma}(-2.5))$Out$-0.056\,243\,716\,497\,674\,054-9.424\,777\,960\,769\,38i$

LogGamma is the continuation: the same real part as GammaLn at $-0.5$, with imaginary part $-\pi$ (Wolfram's LogGamma[-0.5] and mpmath's loggamma(-0.5))

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In$\mathrm{LogGamma}(-0.5)$Out$1.265\,512\,123\,484\,645\,4-3.141\,592\,653\,589\,793i$

Properties #

$\ln\Gamma(\tfrac12) = \tfrac12 \ln\pi$, from $\Gamma(\tfrac12) = \sqrt\pi$

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In$\mathrm{LogGamma}(\frac{1}{2})$Out$\frac{\ln(π)}{2}$

A pole of $\Gamma$

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In$\mathrm{LogGamma}(0)$Out$∞$

On the negative axis: $\Gamma(-\tfrac32) = \tfrac43\sqrt\pi > 0$, but the continuation carries $-2\pi i$

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In$\mathrm{LogGamma}(-1.5)$Out$0.860\,047\,015\,376\,481-6.283\,185\,307\,179\,586i$

Half-integers past $\tfrac12$: $\ln\Gamma(\tfrac32) = \ln\tfrac{\sqrt\pi}{2}$

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In$\mathrm{LogGamma}(\frac{3}{2})$Out$\ln(\frac{\sqrt{π}}{2})$

Possible issues #

Off the positive axis this is the continuation: $\ln(\Gamma(-2.5+1.5i))$ with a principal log has imaginary part $-1.4299$, a full $2\pi$ away

live ↗

In$\mathrm{LogGamma}(-2.5+1.5i)$Out$-3.717\,513\,451\,191\,792\,7-7.713\,065\,525\,834\,192i$

LogGamma(NegativeInfinity) is Indeterminate: the negative axis is LogGamma's own branch cut, where Gamma oscillates in sign between poles with no single limit

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In$\mathrm{LogGamma}(-∞)$Out$\operatorname{Indeterminate}$

Known value · DLMF 5.2, via Gamma(NegativeInfinity) = Indeterminate

About

Examples	25
Checked against	Wolfram Language 24 agreeSageMath 19 agree2 differ2 errorSymPy 20 agree2 differ2 errormpmath 19 agree4 errorJulia 17 agree6 errorRust 9 agree1 differ3 unevaluated6 error
Crosswalk	WikipediaMathWorld ×2WikidataEncyclopedia of MathematicsnLabBritannicaDLMFFungrimWolfram LanguageSageMathSymPympmathJuliaRust

Implementation

referenceenginenotatio

The continuation pinned by lnΓ(1) = 0 and (lnΓ)′ = ψ — which is what makes it the continuation and not Ln(Gamma(z)), whose branch differs off the positive axis.

nativeenginetypescriptpackages/ce-patches/src/compute-engine/numerics/log-gamma.ts

mappedexternalwolfram / mpmathpackages/oracle/src/mappings.ts

LogGamma[z] and mpmath.loggamma(z) — both the continuation, not Log[Gamma[z]].

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

The principal branch, cut along the negative reals, as in Wolfram.

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

See also: Gamma, GammaLn, LogBarnesG, Digamma