(Subscript[E, n])-Function

Author

Eric W. Weisstein
August 3, 2008

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

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

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

Notation

ExpIntegralE[n,x]//TraditionalForm

<<Utilities`Typesetting`

MakeBoxes[En[n_,x_],TraditionalForm]:=RowBox[{SubscriptBox[TagBox[StyleBox["E",FontSlant->"Plain"],ExpIntegralE],MakeBoxes[n,TraditionalForm]],"(",MakeBoxes[x,TraditionalForm],")"}]

En[n,x]//TraditionalForm

Plot

Plot[Evaluate[Table[ExpIntegralE[n,x],{n,0,5}]],{x,0,3},PlotStyle->List/@{Black,Red,Yellow,Green,Blue,Purple},PlotRange->{0,1},
AxesLabel->TraditionalForm/@{x,En[n,x]}]

Complex Plane

<<Utilities`Plot`

ComplexPlot3D[ExpIntegralE[0,z],{z,-5,5},ComplexPlotLabel->En[0,z]]//Timing

ComplexContourPlot[ExpIntegralE[0,z],{z,-5,5},ComplexPlotLabel->En[0,z]]//Timing

ComplexPlot3D[ExpIntegralE[1,z],{z,-5,5},ComplexPlotLabel->En[1,z]]//Timing

Definition

Integrate[((Exp[-x t])/(t^n)),{t,1,∞}]

Integrate[Exp[-x/η](η^(n-2)),{η,0,1}]

Special Values

FunctionExpand[Table[ExpIntegralE[n,x],{n,4}]]

Simplify[FunctionExpand[Table[ExpIntegralE[n,x],{n,4}]],Assumptions->x>0]//ColumnForm//TraditionalForm

FunctionExpand[ExpIntegralE[n,0]]

Table[(0^(-1+n)) Gamma[1-n,0],{n,10}]

Table[ExpIntegralE[n,0],{n,10}]

With[{r=3,dr=.1},
ListDensityPlot[Map[If[NumericQ[#],#,0]&,Table[
Abs[ExpIntegralE[x+I y,0]],{x,-r,r,dr},{y,-r,r,dr}],{2}],
MeshRange->{{-1,1},{-1,1}r},ColorFunction->Hue]
]

Recurrence

D[ExpIntegralE[n,x],x]

Subtract@@{n ExpIntegralE[n+1,x],Exp[-x]-x ExpIntegralE[n,x]}

FullSimplify[%]

FunctionExpand[%]

Table[Subtract@@{n ExpIntegralE[n+1,x],Exp[-x]-x ExpIntegralE[n,x]}/.{x->Random[Complex],n->Random[Complex,5(I+1){-1,1},100]},{100}]//Chop//Union

Table[-Cosh[x]+(x^n) Gamma[1-n,x]+n (x^n) Gamma[-n,x]+Sinh[x]/.{x->Random[Complex],n->Random[Complex]},{100}]//Chop//Union

Special Values

(Subscript[E, n])(0)

Table[ExpIntegralE[n,0],{n,10}]

(Subscript[E, 1])(x)==Γ(0,x)

ExpIntegralE[1,x]//FunctionExpand

Subtract@@@Table[{ExpIntegralE[1,x],Gamma[0,x]}/.x->RandomComplex[2(I+1){-1,1}],{100}]//Chop//Union

Series

(Subscript[E, 1])(x)==-(∑_(n=1)^∞)(((((-1))^n) (x^n))/(n n!))-ln(x)-ℽ==Γ(0,x)

V7

-EulerGamma-Log[x]-Sum[(((((-1))^n)(x^n))/(n! n)),{n,1,∞}]

FullSimplify[Gamma[0,x]]

FunctionExpand[%]

(Subscript[E, n])(x)==exp(-x) (∑_(s=0)^(n-2))(((-x))^s) (n-s-2)!+(((((-x))^(n-1)) (Subscript[E, 1])(x))/((n-1)!))

Series[ExpIntegralE[n,x],{x,0,4}]

sum=(((-x))^(n-1))ExpIntegralE[1,x]+Exp[-x]Sum[(n-s-2)!(((-x))^s),{s,0,n-2}]

FullSimplify[Subtract@@@Table[{sum,ExpIntegralE[n,x]},{n,5}]]

FunctionExpand[%]

Series[ExpIntegralE[n,x],{x,Infinity,4}]//Simplify

Series[ExpIntegralE[n,x],{x,Infinity,4},Assumptions->x>0]//Simplify

V6

(((-x))^(n-1))ExpIntegralE[1,x]+Exp[-x]Sum[(n-s-2)!(((-x))^s),{s,0,n-2}]

V7

(((-x))^(n-1))ExpIntegralE[1,x]+Exp[-x]Sum[(n-s-2)!(((-x))^s),{s,0,n-2}]

FullSimplify[%]

FunctionExpand[%]

(Subscript[E, n])(x)==(∑_(k=0)^∞)(((((-1))^(k+1)) (x^k))/(k! (k-n+1)))+(x^(n-1)) Γ(1-n)

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

(Subscript[E, n])(x)==((exp(-x))/x) (∑_(k=0)^∞)(((((-1))^k) (Subscript[(n), k]))/(x^k))

Table[(((-1))^k)((Pochhammer[n,k])/(x^k)),{k,0,5}]

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

N[With[{x=10^5,n=7},{((Exp[-x])/x)Sum[(((-1))^k)((Pochhammer[1,k])/(x^k)),{k,0,10}],ExpIntegralE[n,x]}],20]