Gamma

The gamma function $\Gamma(z)$, extending the factorial to real and complex arguments; with two arguments, the upper incomplete gamma function.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaGamma function↗MathWorldGammaFunction↗WikidataQ190573↗Encyclopedia of MathematicsGamma-function↗nLabGamma function↗Britannicascience/gamma-function↗DLMF5.2.E1↗FungrimGamma✓↗Fungrim entryWolfram LanguageGamma!↗

Gamma(z)the gamma function $\Gamma(z)$.

SageMathgamma(z)SymPygamma(z)↗mpmathgamma(z)↗Juliaenumeratio_ball(gamma, z)Rustgamma(z)

Gamma(s, z)the upper incomplete gamma function $\Gamma(s, z) = \int_z^\infty t^{s-1} e^{-t}\,dt$.

SageMathgamma_inc(s, z)SymPyuppergamma(s, z)mpmathgammainc(s, z)Juliaenumeratio_ball(gamma, s, z)Rustupper_gamma(s, z)

Gamma(s, z0, z1)the generalized incomplete gamma $\Gamma(s, z_0) - \Gamma(s, z_1) = \int_{z_0}^{z_1} t^{s-1} e^{-t}\,dt$; at $z_0 = 0$ the LOWER incomplete gamma $\gamma(s, z_1)$.

mpmathgammainc(s, z0, z1)Juliaenumeratio_ball((a, z0, z1) -> gamma(a, z0) - gamma(a, z1), s, z0, z1)

Typed signatures

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

Domain: Special functions

Details

The gamma function $\Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\,dt$ for $\operatorname{Re}(z) > 0$ (DLMF 5.2.1), continued analytically elsewhere, is the standard extension of the factorial to real and complex arguments. The two- and three-argument forms below split that same integral at a finite endpoint rather than $\infty$.

• Extends the factorial: $\Gamma(n) = (n-1)!$ for positive integers n, so Factorial is $\Gamma(n+1)$.
• Functional equation: $\Gamma(z+1) = z\,\Gamma(z)$.
• Reflection formula: $\Gamma(z)\,\Gamma(1-z) = \dfrac{\pi}{\sin(\pi z)}$, linking $\Gamma$ at z and $1-z$.
• Poles at the nonpositive integers $0, -1, -2, \ldots$, where $\Gamma$ diverges to ComplexInfinity.
• $\Gamma(1/2) = \sqrt{\pi}$, the constant behind the normal distribution's normalizing factor. See Erf.
• compute-engine's plain evaluation would otherwise leave Gamma at exact integer or rational arguments unevaluated (@enumeratio/analytic overrides it, reducing every integer and half-integer exactly); a floating-point argument, or N(), still reduces to a decimal.
• The three-argument form (@enumeratio/analytic) is Wolfram's generalized incomplete gamma, the integral between two limits: $\Gamma(s, z_0, z_1) = \Gamma(s, z_0) - \Gamma(s, z_1)$. It is the only spelling here for the LOWER incomplete gamma $\gamma(s, z) = \Gamma(s, 0, z)$, which is what the Gamma-distribution CDF and the $\chi^2$ CDF are built from. See GammaRegularized for the normalized version.
• $\Gamma(1, z) = e^{-z}$, also supplied by @enumeratio/analytic -- exact and valid for symbolic $z$, which is what makes $\Gamma(1, 0, z)$ collapse to $1 - e^{-z}$ as Wolfram's does.
• Differs from Wolfram: $Gamma(1/2, x)$ reduces to $\sqrt\pi\,\mathrm{erfc}(\sqrt x)$ on evaluation; Wolfram keeps Gamma[1/2, x] until FunctionExpand. Both agree as identities.

Plot

ComplexPlot

real slice

Examples

13 more kept as data, checked against the oracles

Basic #

A pole of Gamma

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In$Γ(0)$Out$\tilde∞$

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

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In$Γ(2.5)$Out$1.329\,340\,388\,179\,137\,020\,47$

Threads over a list, so poles evaluate concretely even inside one

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In$Γ(\bigl\lbrack0, -1\bigr\rbrack)$Out$\bigl\lbrack\tilde∞, \tilde∞\bigr\rbrack$

Three arguments: the integral between two limits. With $z_0 = 0$ this is the lower incomplete gamma $\gamma(5/2, 3/2)$

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In$Γ(2.5, 0, 1.5)$Out$0.398\,820\,945\,392\,343\,9$

$\int_{3/2}^{3} t^{3/2} e^{-t}\,dt = \Gamma(5/2, 3/2) - \Gamma(5/2, 3)$

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In$Γ(2.5, 1.5, 3)$Out$0.523\,450\,266\,915\,488\,6$

A machine-precision argument: $\Gamma(4.5) = 11.6317\ldots$

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In$Γ(4.5)$Out$11.631\,728\,396\,567\,448\,929\,1$

$\Gamma(5) = 4! = 24$

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In$Γ(5)$Out$24$

Scope #

Threads over a list and reduces each integer argument to the concrete factorial, overriding compute-engine's plain policy of leaving exact Gamma arguments symbolic

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In$Γ(\bigl\lbrack1, 2, 3, 4, 5\bigr\rbrack)$Out$\bigl\lbrack1, 1, 2, 6, 24\bigr\rbrack$

$\Gamma(5/2) = \frac{3}{4}\sqrt{\pi}$, via the half-integer recurrence

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In$Γ(\frac{5}{2})$Out$\frac{3\sqrt{π}}{4}$

Complex arguments evaluate directly

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In$Γ(2.3+i)$Out$0.719\,140\,936\,537\,281\,1+0.540\,614\,467\,909\,848\,7i$

Half-integers have closed forms: $\Gamma(7/2) = \frac{15}{8}\sqrt{\pi}$

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In$Γ(\frac{7}{2})$Out$\frac{15\sqrt{π}}{8}$

...including negative half-integers: $\Gamma(-3/2) = \frac{4}{3}\sqrt{\pi}$

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In$Γ(-\frac{3}{2})$Out$\frac{4\sqrt{π}}{3}$

$\Gamma(\infty) = \infty$

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In$Γ(∞)$Out$∞$

Values far past the double range: $\Gamma(200.5) \approx 5.57 \times 10^{373}$

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In$Γ(200.5)$Out$5.573\,168\,944\,801\,379\,133\,64\cdot10^{373}$

Two arguments: the upper incomplete gamma $\Gamma(5/2, 3/2)$

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In$Γ(2.5, 1.5)$Out$0.930\,519\,442\,786\,793\,1$

An integer order reduces to a closed form: $\Gamma(2, x) = (1 + x)e^{-x}$

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In$Γ(2, x)$Out$(x+1)e^{-x}$

$\Gamma(2, 1) = 2/e$, the same closed form at an exact argument

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In$Γ(2, 1)$Out$\frac{2}{e}$

Listable: threads elementwise over a matrix, and each entry reduces ($\Gamma(2, 0) = 1$)

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In$Γ(2, \bigl\lbrack\bigl\lbrack\frac{7}{2}, 0\bigr\rbrack, \bigl\lbrack0, \frac{13}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{9}{2e^{7/2}}, 1\bigr\rbrack, \bigl\lbrack1, \frac{15}{2e^{13/2}}\bigr\rbrack\bigr\rbrack$

Interval arithmetic: the image of $[1.4, 1.5]$, whose lower end is $\Gamma$'s minimum at $x_0 = 1.4616\ldots$

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In$Γ(\mathrm{Interval}(1.4, 1.5))$Out$\mathrm{Interval}(0.885\,603\,194\,410\,888\,7, 0.887\,263\,817\,503\,075\,3)$

Uncertainty propagation: $\Gamma(2.5 \pm 0.01) = 1.3293 \pm 0.0093$

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In$Γ(\mathrm{Around}(2.5, 0.01))$Out$\mathrm{Around}(1.329\,340\,388\,179\,137, 0.009\,347\,345\,216\,260\,856)$

To 40 significant digits, every one correctly rounded

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In$N(Γ(\frac{1}{3}), 40)$Out$2.678\,938\,534\,707\,747\,633\,655\,692\,940\,974\,677\,644\,129$

$\Gamma(-\infty)$ is Indeterminate -- the poles at every non-positive integer accumulate along the negative real axis

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

Known value · Wolfram Gamma[-Infinity]

$\Gamma(\tilde\infty)$ is Indeterminate: no direction to take the limit along

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In$Γ(\tilde∞)$Out$\operatorname{Indeterminate}$

$\lim_{t\to\infty}\Gamma(it) = 0$: $|\Gamma(iy)| \to 0$ as $y \to \infty$

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In$\lim_{t→∞}Γ(i t)$Out$0$

Known value · DLMF 5.11.9; Wolfram Gamma[I Infinity]

Gamma of the directed infinity $i\infty$ stays unevaluated; the limit along it is in the Limit example above

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In$Γ(\mathrm{DirectedInfinity}(i))$Out$Γ(\mathrm{DirectedInfinity}(i))$

Known value · DLMF 5.11.9; Wolfram Gamma[I Infinity]

Threads over a list of infinities, each reducing to its own value

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In$\mathrm{Map}(Γ, \bigl\lbrack∞, -∞, \tilde∞\bigr\rbrack)$Out$\bigl\lbrack∞, \operatorname{Indeterminate}, \operatorname{Indeterminate}\bigr\rbrack$

Applications #

The volume of a unit 4-ball is $\pi^{n/2}/\Gamma(n/2+1)$; at $n=4$ that's $\pi^2/\Gamma(3) = \pi^2/2$

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In$\frac{π^2}{Γ(3)}=\frac{π^2}{2}$Out$\mathrm{True}$

Properties #

Extends the factorial: $\Gamma(n+1) = n!$, so $\Gamma(6) = 5!$

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In$Γ(6)=5!$Out$\mathrm{True}$

Functional equation $\Gamma(z+1) = z\,\Gamma(z)$, here at $z = 5$

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In$Γ(6)=5Γ(5)$Out$\mathrm{True}$

Reflection formula $\Gamma(z)\Gamma(1-z) = \pi/\sin(\pi z)$, here at $z = 1/3$

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In$Γ(\frac{1}{3})Γ(\frac{2}{3})=\frac{π}{\sin(\frac{π}{3})}$Out$\mathrm{True}$

$\Gamma(1/2) = \sqrt{\pi}$

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In$Γ(\frac{1}{2})=\sqrt{π}$Out$\mathrm{True}$

$\Gamma(1, z) = e^{-z}$, exact and symbolic in $z$

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In$Γ(1, z)$Out$e^{-z}$

…so $\gamma(1, z) = 1 - e^{-z}$: the exponential distribution's CDF, falling out of the general form

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In$Γ(1, 0, z)$Out$1-e^{-z}$

Now that $\Gamma(2, z)$ has a closed form, the three-argument difference reduces through it too: $\Gamma(2, 0, z) = 1 - (1+z)e^{-z}$. Wolfram's bare Gamma[2, 0, z] stays unevaluated (only FunctionExpand reaches this); this engine reduces both

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In$Γ(2, 0, z)$Out$1-(z+1)e^{-z}$

$\Gamma(a, 0) = \Gamma(a)$: the incomplete gamma from zero is the complete one

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In$Γ(a, 0)$Out$Γ(a)$

$\Gamma(0, z) = E_1(z)$, the exponential integral: $E_1(3/2) = 0.10002\ldots$

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In$Γ(0, 1.5)$Out$0.100\,019\,582\,406\,632\,52$

$\Gamma(1/2, x) = \sqrt{\pi}\,\operatorname{erfc}(\sqrt{x})$. See Erfc

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In$Γ(\frac{1}{2}, x)$Out$\operatorname{erfc}(\sqrt{x})\sqrt{π}$

$\Gamma'(x) = \Gamma(x)\,\psi(x)$, so the logarithmic derivative is Digamma

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

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In$\frac{1}{Γ(\frac{1}{2})}$Out$\sqrt{\frac{1}{π}}$

Known value · Wolfram MathematicalFunctionData, Gamma, Euler's infinite product

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In$Γ(\frac{1}{2})^2$Out$π$

Known value · Wolfram MathematicalFunctionData, Gamma, Euler's reflection formula

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In$Γ(-\frac{1}{2})Γ(\frac{1}{2})$Out$-2π$

Known value · Wolfram MathematicalFunctionData, Gamma, Reflection formula

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In$\prod_{k=0}^1Γ(\frac{1}{2}(2k+\frac{1}{2}))$Out$Γ(\frac{1}{4})Γ(\frac{5}{4})$

Known value · Wolfram MathematicalFunctionData, Gamma, Schobloch's reciprocity formula

Neat examples #

Squaring the reflection formula at $z=1/2$ collapses it to $\Gamma(1/2)^2 = \pi$

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In$Γ(1/2)^2=π$Out$\mathrm{True}$

$|\Gamma(i)|^2 = \pi/\sinh\pi = 0.27203\ldots$

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In$N(\vert Γ(i)\vert^2)$Out$0.272\,029\,054\,982\,132\,823\,542$

About

Examples	57
Checked against	Wolfram Language 52 agree4 differ1 unevaluatedSageMath 49 agree3 differSymPy 50 agree2 differmpmath 38 agree4 errorJulia 35 agree7 errorRust 19 agree2 unevaluated5 error
Crosswalk	WikipediaMathWorldWikidataEncyclopedia of MathematicsnLabBritannicaDLMFFungrimFungrim entry ×101Wolfram LanguageSageMath ×2SymPy ×2mpmath ×3Julia ×3Rust ×2

Implementation

mappedwolframpackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

Wolfram's generalized incomplete gamma is the difference of two upper incomplete gammas, written out; at z0 = 0 it is the lower incomplete gamma.

mappedmpmathpackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

The incomplete gamma from z0 to z1, the difference of two upper ones.

mappedsagepackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

a, z > 0 only.

See also: Factorial, GammaLn, LogGamma, GammaRegularized, Digamma, Beta