Proof of Equation (4.1)
\begin{equation*}
{{E}_{(1)}}\left( x \right) =\int_{x}^{\infty }{{{t}^{-1}}{\exp(-t)}dt}.
\end{equation*}
Proof:
Let
u=xt,
du=xdt, and
t=u/x. We transform
{{E}_{(1)}} into an integral function of
u
\begin{align*}
{{E}_{(1)}}\left( x \right) &=\int_{1}^{\infty }{\frac{\exp (-xt)}{t}dt} \\
& =\int_{x}^{\infty }{{{u}^{-1}}\exp (-u)du}.
\end{align*}
We can change the variable
u back to
t and conclude the proof.
\square