GammaLn

The natural logarithm of $\Gamma(z)$, avoiding the overflow of computing Gamma directly for large z.

✓ 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↗Fungrim entryWolfram LanguageLogGamma!↗

GammaLn(z)$\ln \Gamma(z)$, useful where $\Gamma(z)$ itself would overflow.

SageMathlog(gamma(z))!↗SymPylog(gamma(z))↗mpmathlog(gamma(z))↗Juliaenumeratio_ball(z -> log(gamma(z)), 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

• Defined via Gamma: $\operatorname{GammaLn}(z) = \ln \Gamma(z)$ for $z > 0$, where $\Gamma$ is positive so no branch-cut ambiguity arises.
• Called gammaln in MATLAB and SciPy, lgamma in C's math library.
• Inherits Gamma's recurrence in log form: $\operatorname{GammaLn}(z+1) = \operatorname{GammaLn}(z) + \ln z$.
• $\operatorname{GammaLn}(1/2) = \frac{1}{2}\ln \pi$, from $\Gamma(1/2) = \sqrt{\pi}$.
• Diverges to $+\infty$ at the nonpositive integers, the poles of Gamma -- the log of a diverging magnitude, rather than the ComplexInfinity that Gamma itself returns there.
• GammaLn is the real log-magnitude, not the complex log: for real $z<0$ it is $\ln|\Gamma(z)|$, so $\operatorname{GammaLn}(-0.5) = \ln(2\sqrt\pi) \approx 1.2655$ with no imaginary part, though $\Gamma(-0.5) = -2\sqrt\pi < 0$. LogGamma is the principal continuation, $\ln\Gamma(z)$ analytic off $(-\infty, 0]$: $\operatorname{LogGamma}(-0.5) \approx 1.2655 - \pi i$ (Wolfram's LogGamma[-0.5] and mpmath's loggamma(-0.5), both $1.2655 - 3.1416i$; the principal logarithm of the negative real $\Gamma(-0.5)$ itself would be $+\pi i$, so it is a third value). Use GammaLn for the magnitude of a real gamma ratio, LogGamma where continuity in $z$ matters.
• For complex $z$ it is $\ln(\Gamma(z))$ with a PRINCIPAL logarithm, which is not Wolfram's $\mathrm{LogGamma}$: the two differ by multiples of $2\pi i$ off the positive axis.

Plot

ComplexPlot

real slice

Examples

6 more kept as data, checked against the oracles

Basic #

A pole of Gamma, in log form

live ↗

In$\mathrm{GammaLn}(0)$Out$∞$

A floating-point argument evaluates directly to a high-precision decimal

live ↗

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

$\ln\Gamma(100.5)$, where $\Gamma(100.5)$ itself is about $10^{157}$

live ↗

In$\mathrm{GammaLn}(100.5)$Out$361.435\,540\,467\,777\,621\,555$

Scope #

At a pole of Gamma, compute-engine gives the real log-magnitude's divergence, PositiveInfinity, rather than treating GammaLn as complex-analytic

live ↗

In$\mathrm{GammaLn}(-1)$Out$∞$

Complex arguments; here the principal $\ln\Gamma(z)$ and Wolfram's $\mathrm{LogGamma}$ agree, as the imaginary part stays inside $(-\pi, \pi]$

live ↗

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

A huge inexact argument, far past a double, from Stirling's series: $\ln\Gamma(10^{1000}) \approx 2.3016 \times 10^{1003}$

live ↗

In$\mathrm{GammaLn}(1.0\cdot10^{1000})$Out$230\,158\,509\,299\,404\,568\,402\cdot10^{983}$

Threads over a list, and the integer points reduce to $\ln((n-1)!)$, matching Wolfram's LogGamma[5] = Log[24]

live ↗

In$\mathrm{GammaLn}(\bigl\lbrack1, 2, 3\bigr\rbrack)$Out$\bigl\lbrack0, 0, \ln(2)\bigr\rbrack$

Uncertainty propagation through $\ln\Gamma$, with slope $\psi(1.2)$

live ↗

In$\mathrm{GammaLn}(\mathrm{Around}(1.2, 0.01))$Out$\mathrm{Around}(-0.085\,374\,090\,003\,315\,83, 0.002\,890\,398\,965\,921\,884)$

To 30 significant digits, the last one correctly rounded

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

GammaLn is the real log-magnitude $\ln|\Gamma(-0.5)| = \ln(2\sqrt\pi)$, with no imaginary part where $\Gamma<0$

live ↗

In$\mathrm{GammaLn}(-0.5)$Out$1.265\,512\,123\,484\,645\,396\,49$

Applications #

Used to get $\ln(n!)$ for large n without overflowing float64: $\ln(10!) = \operatorname{GammaLn}(11)$

live ↗

In$\mathrm{GammaLn}(11)=\ln(10!)$Out$\mathrm{True}$

Properties #

Defined as $\ln \Gamma(z)$, here at $z = 5$

live ↗

In$\mathrm{GammaLn}(5)=\ln(Γ(5))$Out$\mathrm{True}$

$\operatorname{GammaLn}(1) = \ln \Gamma(1) = \ln 1 = 0$

live ↗

In$\mathrm{GammaLn}(1)=0$Out$\mathrm{True}$

Inherits Gamma's recurrence: $\operatorname{GammaLn}(z+1) = \operatorname{GammaLn}(z) + \ln z$

live ↗

In$\mathrm{GammaLn}(6)=\mathrm{GammaLn}(5)+\ln(5)$Out$\mathrm{True}$

$\operatorname{GammaLn}(1/2) = \frac12 \ln \pi$, from $\Gamma(1/2) = \sqrt{\pi}$

live ↗

In$\mathrm{GammaLn}(\frac{1}{2})=\frac{\ln(π)}{2}$Out$\mathrm{True}$

An exact positive integer now reduces too, matching Wolfram's LogGamma[5] = Log[24] ($\ln 24 = 3\ln 2 + \ln 3$, compute-engine's own factored form for $\ln$ of a composite)

live ↗

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

The derivative of $\ln\Gamma$ is the digamma function. See Digamma

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In$\frac{\mathrm{d}\mathrm{GammaLn}(x)}{\mathrm{d}x}$Out$\mathrm{Digamma}(x)$

Possible issues #

Off the positive axis GammaLn is the principal $\ln\Gamma(z)$: at $z = -3/2$, where $\Gamma > 0$, that is real. See LogGamma for the continuation

live ↗

In$\mathrm{GammaLn}(-1.5)$Out$0.860\,047\,015\,376\,481\,014\,511$

Neat examples #

$\operatorname{GammaLn}(101)$ is exactly $\ln(100!)$, letting us reason about the size of the 158-digit number $100!$ without computing it

live ↗

In$\mathrm{GammaLn}(101)=\ln(100!)$Out$\mathrm{True}$

About

Examples	25
Checked against	Wolfram Language 20 agree4 differSageMath 20 agree1 differ2 unevaluatedSymPy 19 agree2 differ3 unevaluatedmpmath 17 agree3 unevaluated3 errorJulia 17 agree1 differ3 unevaluated2 errorRust 10 agree3 differ5 error
Crosswalk	WikipediaMathWorld ×2WikidataEncyclopedia of MathematicsnLabBritannicaDLMFFungrimFungrim entry ×16Wolfram LanguageSageMathSymPympmathJuliaRust

Implementation

mappedwolframpackages/oracle/src/mappings.ts

The principal logarithm of Gamma, not LogGamma, which is the continuation.

mappedsympypackages/oracle/src/mappings.ts

The principal logarithm of Gamma, not SymPy's loggamma (the continuation, which is LogGamma here). At a negative real where Gamma is negative it adds pi i to the real log-magnitude GammaLn takes.

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

See also: Gamma, LogGamma, Digamma