Gamma

The gamma function , extending the factorial to real and complex arguments; with two arguments, the upper incomplete gamma function.

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Gamma(z)the gamma function .

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Gamma(s, z)the upper incomplete gamma function .

Gamma(s, z0, z1)the generalized incomplete gamma ; at the LOWER incomplete gamma .

Domain: Special functions

Details

Extends the factorial: for positive integers n, so Factorial is .
Functional equation: .
Reflection formula: , linking at z and .
Poles at the nonpositive integers , where diverges to ComplexInfinity.
, the constant behind the normal distribution's normalizing factor. See Erf.
compute-engine's plain evaluation leaves Gamma at exact integer or rational arguments unevaluated -- it only reduces to a decimal with N() or when given an inexact (floating-point) argument.
The three-argument form (`@enumeratio/analytic`) is Wolfram's generalized incomplete gamma, the integral between two limits: . It is the only spelling here for the LOWER incomplete gamma , which is what the Gamma-distribution CDF and the CDF are built from. See GammaRegularized for the normalized version.
, also supplied by `@enumeratio/analytic` -- exact and valid for symbolic , which is what makes collapse to as Wolfram's does.

Examples