GammaRegularized

The regularized upper incomplete gamma function , the survival function of the Gamma(a, 1) distribution.

Wolfram LanguageGammaRegularized!

GammaRegularized(a, z)the regularized upper incomplete gamma .

GammaRegularized(a, z0, z1)the regularized difference ; at the LOWER regularized .

Domain: Special functions

Details

Complementary to the lower regularized , which is the three-argument (`@enumeratio/analytic`) -- the Gamma-distribution CDF, and with , the CDF.
for any : all the mass sits to the right of 0.
For a positive integer n it has a closed polynomial form, , which compute-engine uses to evaluate exactly, even for negative z.
Gives the Poisson CDF tail and the Gamma-distribution survival function, so it shows up throughout reliability and queueing models. See Gamma.
compute-engine leaves a non-integer a combined with negative z symbolic -- genuinely outside the real domain there -- rather than erroring or returning NaN.
The three-argument form divides the difference by rather than subtracting two values, so holds for negative too, where and are separately infinite but their ratio is 1. That is Wolfram's convention.
A genuinely complex or , which the built-in handler declines, is evaluated as (`@enumeratio/analytic`) -- Gamma itself takes complex arguments.

Examples