ExpIntegralEi

({{, ExpIntegralEi[z]
gives the exponential integral function Ei(z). }})

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

Ei(z)=-(∫_(-z)^∞)(e^(-t))/tdt, where the principal value of the integral is taken.

ExpIntegralEi[z] has a branch cut discontinuity in the complex z plane running from -∞ to 0.

For certain special arguments, ExpIntegralEi automatically evaluates to exact values.

ExpIntegralEi can be evaluated to arbitrary numerical precision.

ExpIntegralEi automatically threads over lists.

EXAMPLES

Basic Examples  (3)

Evaluate numerically:

ExpIntegralEi[1.8]

Plot[ExpIntegralEi[x],{x,-1,1}]

Series expansion around the branch point at the origin:

Series[ExpIntegralEi[x],{x,0,5},Assumptions->x>0]

Series[ExpIntegralEi[x],{x,0,5},Assumptions->x<0]

Scope  (5)

Evaluate to high precision:

N[ExpIntegralEi[2],50]

The precision of the output tracks the precision of the input:

ExpIntegralEi[2.0000000000000000000000]

ExpIntegralEi threads element-wise over lists:

ExpIntegralEi[{1.2,1.5,1.8}]

ExpIntegralEi can take complex number inputs:

ExpIntegralEi[2.+I]

Simple exact values are generated automatically:

ExpIntegralEi[0]

ExpIntegralEi[Infinity]

TraditionalForm formatting:

ExpIntegralEi[x]//TraditionalForm

Generalizations & Extensions  (2)

ExpIntegralEi can be applied to power series:

ExpIntegralEi[1+x+((x^2)/2)+((x^3)/9)+((O[x])^4)]

Find series expansions at infinity:

Series[ExpIntegralEi[x],{x,Infinity,3}]

Give the result for an arbitrary symbolic direction:

Series[ExpIntegralEi[x],{x,DirectedInfinity[],1}]

Applications  (2)

Compute a classical asymptotic series with k! coefficients:

Series[-Exp[1/z] ExpIntegralEi[-1/z],{z,0,10},Assumptions->z>0]

Plot the imaginary part in the complex plane:

Plot3D[Im[ExpIntegralEi[x+I y]],{x,-3,3},{y,-3,3}]

Properties & Relations  (8)

Use FullSimplify to simplify expressions containing exponential integrals:

FullSimplify[ExpIntegralEi[z]+(1/2)(Log[z]+Log[(1/z)])]

Find the numerical root:

FindRoot[ExpIntegralEi[z]==0,{z,1}]

Obtain ExpIntegralEi from integrals and sums:

Integrate[(E^t-1)/t,{t,0,z},Assumptions->z>0]

(∑_(k=1)^∞)((z^k)/(k k!))//FunctionExpand

Calculate limits:

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

Obtain ExpIntegralEi from a differential equation:

DSolve[z Derivative[3][w][z]+2 w''[z]-z w'[z]==0,w,z]

Calculate Wronskian:

Det[Outer[D[#1,{x,#2}]&,{1,ExpIntegralEi[x],ExpIntegralEi[-x]},{0,1,2}]]

Integrals:

∫(z^α)ExpIntegralEi[(z^β)]\[DifferentialD]z

Integral transforms:

LaplaceTransform[ExpIntegralEi[t],t,s,Assumptions->s>1]

Possible Issues  (3)

ExpIntegralEi can take large values for moderate-size arguments:

ExpIntegralEi[Sqrt[I] 10000. ]

ExpIntegralEi has a special value on the negative real axis, not obtained as a limit from either side:

N[ExpIntegralEi[-1+{-10^-10I,0,10^-10 I}]]

A larger setting for $MaxExtraPrecision can be needed:

N[ExpIntegralEi[-10^60 I]+ I Pi,20]

Block[{$MaxExtraPrecision=200},N[ExpIntegralEi[-10^60 I]+ I Pi,20]]

Neat Examples  (1)

Nested integrals:

NestList[Integrate[#,x]&,ExpIntegralEi[ x],4]

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