(Subscript[E, t])-Function

Author

Eric W. Weisstein
February 17, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/Et-Function.nb.

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

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

Definition

Et[t_,ν_,a_]:=(((a^(-ν)) (E^(a t)) Gamma[ν,0,a t])/(Gamma[ν]))

Definition

expr1=((Exp[a t])/(Gamma[ν]))Integrate[(x^(ν-1))Exp[-a x],{x,0,t},Assumptions->Re[ν]>0&&t>0]

expr1 /. (Gamma[x_]-Gamma[x_,y_]:>Gamma[x,0,y])//TraditionalForm

Recurrance Relation

FullSimplify[Subtract@@{Et[t,ν,a],a Et[t,ν+1,a]+((t^ν)/(Gamma[ν+1]))},t>0]

Special Value

Limit[Et[t,ν,a],ν->0]