File:AsymptoticExpansionE1.png

Divergence of the asymptotic series for the w:Exponential integral

${\displaystyle {\rm {E}}_{1}(z)=\int _{1}^{\infty }{\frac {e^{-tz}}{t}}\,{\rm {d}}t,\qquad \Re (z)\geq 0~~.}$

The function

${\displaystyle E_{1}(z,N)={\frac {\exp(-z)}{z}}\sum _{n=0}^{N-1}{\frac {n!}{(-z)^{n}}}}$

gives the first N terms of an approximation to ${\displaystyle E_{1}(z)}$ for ${\displaystyle ~{\rm {Re}}(z)\gg 1}$.

The relative error

${\displaystyle {\frac {E_{1}(x,N)-E_{1}(x)}{E_{1}(x)}}}$

is plotted versus ${\displaystyle ~x~}$ for ${\displaystyle 3<x<9}$ for various numbers of terms, N, as follows:
${\displaystyle ~N=1~}$ (red),
${\displaystyle ~N=2~}$ (green),
${\displaystyle ~N=3~}$ (yellow),
${\displaystyle ~N=4~}$ (blue),
${\displaystyle ~N=5~}$ (pink).

${\displaystyle ~x~}$