Regularized Gamma Function

Author

Eric W. Weisstein
December 17, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/RegularizedGammaFunction.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/RegularizedGammaFunction.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Definitions

γ[a_,x_]:=Gamma[a,0,x]
Γ[a_,x_]:=Gamma[a,x]

GammaP[a_,z_]:=GammaRegularized[a,0,z]
GammaQ[a_,z_]:=GammaRegularized[a,z]

P(a,z)

FullSimplify[FunctionExpand[γ[a,z]/Gamma[a]]]

FunctionExpand[GammaRegularized[a,0,z]]

GammaRegularized[a,0,z]//TraditionalForm

FunctionExpand[γ[a,z]/Gamma[a]-GammaP[a,z]]

Derivatives

FullSimplify[Table[D[GammaP[a,z],{z,n}],{n,3}]]

Integral

Integrate[GammaP[a,z],z]

FullSimplify[FunctionExpand[Integrate[GammaP[a,z],z]]]

Q(a,z)

FunctionExpand[Γ[a,z]/Gamma[a]]

FunctionExpand[GammaRegularized[a,z]]

GammaRegularized[a,z]//TraditionalForm

FunctionExpand[Γ[a,z]/Gamma[a]-GammaQ[a,z]]

Derivatives

FullSimplify[Table[D[GammaQ[a,z],{z,n}],{n,3}]]

Integral

Integrate[GammaQ[a,z],z]

FullSimplify[FunctionExpand[Integrate[GammaQ[a,z],z]]]

Identities

GammaP(a,z)+GammaQ(a,z)==1

FullSimplify[GammaP[a,z]+GammaQ[a,z]]