Incomplete Gamma Function

Author

Eric W. Weisstein
August 27, 2008

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

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

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

Upper Γ(a,x)

Integrals

Integrate[t^(a-1)Exp[-t],{t,0,∞},Assumptions->Re[a]>0]

Integrate[t^(a-1)Exp[-t],{t,x,∞}]

Sum

V6

FullSimplify[(n-1)!Exp[-x]Sum[x^s/s!,{s,0,n-1}]]

V7

FullSimplify[(n-1)!Exp[-x]Sum[x^s/s!,{s,0,n-1}]]

Continued Fraction

Integrate[((Exp[-u])/u),{u,x,∞}]

With[{x=1.1},{((Exp[-x])/(x+1+ContinuedFractionK[-n^2,x+2n+1,{n,20}])),Gamma[0,x]}]

((Exp[-x])/(x+1+ContinuedFractionK[-n^2,x+2n+1,{n,∞}]))

Lower Γ(a,0,x)

Integral

Integrate[t^(a-1)Exp[-t],{t,0,x},Assumptions->{Re[a]>0,x>0}]

In terms of Hypergeometrics

FullSimplify[FunctionExpand[x^a/a Exp[-x]Hypergeometric1F1[1,1+a,x]]]

FullSimplify[FunctionExpand[x^a/a Hypergeometric1F1[a,1+a,-x]]]

As a Sum

V6

FullSimplify[(n-1)!(1-Exp[-x]Sum[x^k/k!,{k,0,n-1}])]

V7

FullSimplify[(n-1)!(1-Exp[-x]Sum[x^k/k!,{k,0,n-1}])]

FullSimplify[%]

In terms of other Gamma Functions

FunctionExpand[Gamma[a,0,x]]

FunctionExpand[Gamma[a,s,x]]

Identity

FullSimplify[Gamma[a,0,x]-(Gamma[a]-Gamma[a,x])]

In terms of ei(z)

FunctionExpand[Gamma[0,z]]

FullSimplify[Gamma[0,x]-(-ExpIntegralEi[-x]-I Pi),x<0]

FullSimplify[Gamma[0,x]-(-ExpIntegralEi[-x]),x>0]