Exponential Integral

Author

Eric W. Weisstein
January 6, 2006

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

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

©2006 Wolfram Research, Inc. except for portions noted otherwise

Definition

-(∫_(-x)^∞)((e^(-t))/t)\[DifferentialD]t

-Integrate[((Exp[-t])/t),{t,-x,∞},PrincipalValue->True]

Integrate[((Exp[-u])/u),{u,x,∞},Assumptions->x>0]

Integrate[((Exp[-u])/u),{u,x,∞},Assumptions->x<0]

Plot

<<Utilities`Plot`

Real Axis

PlotDiscontinuous[ExpIntegralEi[x],{x,-2,0,3},
VerticalAsymptotes->{0},
HorizontalAsymptotes->{0},
PlotStyle->Red,
AxesLabel->TraditionalForm/@{x,ExpIntegralEi[x]}
]

Plots in the Complex Plane

ComplexPlot3D[ExpIntegralEi[z],{z,-2,2}]//Timing

ComplexContourPlot[ExpIntegralEi[z],{z,-5,5}]//Timing

Special Values

Gompertz Constant

FunctionExpand[-E ExpIntegralEi[-1]]

N[-E ExpIntegralEi[-1],20]

Ei(1)

N[ExpIntegralEi[1],10]

Root

Solve[ExpIntegralEi[x]==0,x]

root=x/.FindRoot[ExpIntegralEi[x],{x,.01,1},WorkingPrecision->100]

Log[x/.FindRoot[LogIntegral[x],{x,1.2,2},WorkingPrecision->100]]

v={1,Pi,Pi^2,E,EulerGamma,Catalan,Zeta[3],root};
LatticeReduce[Rationalize[N[Transpose[Join[IdentityMatrix[Length[v]],
{10^15v}]],30],10^-30]]//N//MatrixForm

Limit

Limit[Exp[2ExpIntegralEi[-x]]/x^2,x->0]

N[%,20]

RealDigits[%]

Series

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

FullSimplify[FunctionExpand[ExpIntegralEi[x]-Gamma[0,x]],x>0]

Integrals

(∫_1^∞)((exp(-t x))/t)dt

Integrate[Exp[-t x]/t,{t,1,∞}]

(∫_x^∞)((exp(-u))/u)du

Integrate[Exp[-u]/u,{u,x,∞}]

-(∫_(-x)^∞)((exp(-t))/t)dt

-Integrate[Exp[-t]/t,{t,-x,∞}]

(∫_x^∞)((exp(-t))/t)dt

Integrate[t^(-1)Exp[-t],{t,x,∞}]

(∫_(1/2)^∞)((exp(-t))/t)dt

Integrate[t^(-1)Exp[-t],{t,1/2,∞}]

(∫_(-(1/2))^∞)((exp(-t))/t)dt

Integrate[t^(-1)Exp[-t],{t,-1/2,∞}]

Tables

Table[{-ExpIntegralEi[-x],Gamma[0,x]},{x,-1,1,.1}]

In terms of other functions

EulerGamma+Log[z]+Sum[(1/(k k!))(z^k),{k,∞}]

FunctionExpand[%]

In terms of Γ(0,z)

FunctionExpand[Gamma[0,z]]

In Terms of ci and si

FunctionExpand[ExpIntegralEi[I x]]

Simplify[%,x∈Reals]//Expand

Simplify[%,x<0]//Expand

Simplify[%,x>0]//Expand

Subtract@@@Table[{ExpIntegralEi[I x],((I π)/2)+CosIntegral[x]+I SinIntegral[x]}/.x->Random[Real,{0,100}],{20}]//Chop//Union

Particular Values of x

FullSimplify[ExpIntegralEi[-x],x>0]

FullSimplify[-ExpIntegralEi[-x],x>0]

FullSimplify[-ExpIntegralEi[x],x>0]

FullSimplify[ExpIntegralEi[x],x>0]

FullSimplify[ExpIntegralEi[x],x<0]

FullSimplify[ExpIntegralEi[-x],x<0]

FullSimplify[-ExpIntegralEi[-x],x<0]

FunctionExpand[-ExpIntegralEi[-x],x>0]

Series

v5.0

Series[ExpIntegralEi[z],{z,0,10}]

Series[ExpIntegralEi[-z],{z,0,10}]

(EulerGamma+Log[z])+Plus@@Table[((((-1))^k)/(k k!))(z^k),{k,9}]

v5.1

Series[ExpIntegralEi[z],{z,0,10}]

Series[ExpIntegralEi[z],{z,0,10},Assumptions->z>0]

Series[ExpIntegralEi[-z],{z,0,10},Assumptions->z>0]

(EulerGamma+Log[z])+Plus@@Table[((((-1))^k)/(k k!))(z^k),{k,9}]