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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

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