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Regularized Gamma Function

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The regularized gamma functions are defined by

P(a,z)=(gamma(a,z))/(Gamma(a))
(1)
Q(a,z)=(Gamma(a,z))/(Gamma(a)),
(2)

where gamma(a,z) and Gamma(a,z) are incomplete gamma functions and Gamma(a) is a complete gamma function. The function P(a,z) is implemented in the Wolfram Language as GammaRegularized[a, 0, z], and Q(a,z) is implemented as GammaRegularized[a, z].

P(a,z) and Q(a,z) satisfy the identity

P(a,z)+Q(a,z)=1.
(3)

The derivatives of P(a,z) and Q(a,z) are

d/(dz)P(a,z)=(e^(-z)z^(a-1))/(Gamma(a))
(4)
d/(dz)Q(a,z)=-(e^(-z)z^(a-1))/(Gamma(a)),
(5)

and the second derivatives are

(d^2)/(dz^2)P(a,z)=(e^(-z)(a-z-1)z^(a-2))/(Gamma(a))
(6)
(d^2)/(dz^2)Q(a,z)=(e^(-z)(1+z-a)z^(a-2))/(Gamma(a)).
(7)

The integrals are

intP(a,z)dz=(zGamma(a)-zGamma(a,z)+Gamma(a+1,z))/(Gamma(a))
(8)
intQ(a,z)dz=(zGamma(a,z)-Gamma(a+1,z))/(Gamma(a)).
(9)

See also

Gamma Function, Incomplete Gamma Function, Regularized Beta Function

Related Wolfram sites

http://functions.wolfram.com/GammaBetaErf/GammaRegularized/, http://functions.wolfram.com/GammaBetaErf/GammaRegularized3/

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References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 160-161, 1992.

Referenced on Wolfram|Alpha

Regularized Gamma Function

Cite this as:

Weisstein, Eric W. "Regularized Gamma Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RegularizedGammaFunction.html

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