General Calculations

ExpIntegralEi[-0.2]//N

ExpIntegralEi[-0.5]//N

ExpIntegralEi[-1]//N

ExpIntegralEi[0.5]//N

ExpIntegralEi[1]//N

ExpIntegralEi[10]//N

ExpIntegralEi[-10]//N

Plot[-ExpIntegralEi[-x],{x,0,10}]

LogPlot[-ExpIntegralEi[-x],{x,0,10}]

Series[ExpIntegralEi[x],{x,0,10}, Assumptions->Element[x,Reals]]

a1=Series[ExpIntegralEi[x],{x,0,10}, Assumptions->Element[x,Reals]]//Normal

2 Pi I //ComplexExpand

f1[x_]:=EulerGamma+Log[Abs[x]] + x+((x^2)/4)+((x^3)/18)+((x^4)/96)+((x^5)/600)+((x^6)/4320)+((x^7)/35280)+((x^8)/322560)+((x^9)/3265920)+((x^10)/36288000)

Plot[-f1[-x]//N,{x,0,10}]

Plot[(-f1[-x]//N)-(-ExpIntegralEi[-x]),{x,0,10}]

Log[10^-10]//N

For Class

For the Exponential Integral defined by :

Ei[x]=-(∫_(-x)^∞)((Exp[-u])/u)du

For desired precision ϵ  (for example (10^(-6)), (10^(-10)), (10^(-20)))

If x < Abs[Ln[ϵ]], use

γ + Ln[x] + (∑_(n=1)^∞)((x^n)/(n*n!)) until ((x^n)/(n*n!))<ϵ

If x >= Abs[Ln[ϵ]]

((Exp[x])/x)( (∑_(n=1)^∞)((n!)/(x^n)) until ((n!)/(x^n))<ϵ)

Where γ is Euler's constant:

N[EulerGamma,100]