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The Wikipedia entry on Asymptotic Expansion outlines a detailed example, where it refers to the fact that the integral \begin{equation} \int_0^\infty \frac{e^{-w/t}}{1-w} \, dw \end{equation} understood as a Cauchy Principal value, can be expressed in terms of the exponential integral \begin{equation} \text{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} \, dt \end{equation} in the form \begin{equation} \int_0^\infty \frac{e^{-w/t}}{1-w} \, dw = e^{-1/t} \, \text{Ei}\left(\frac{1}{t}\right) \end{equation}

I tried to fill in the details for myself, using various changes of variables, but somehow I manage to make a mistake each time because I cannot get to the desired form. How can I see the connection between these integrals ?

Many thanks for your help!!

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up vote 7 down vote accepted

Changing the variable from $t$ to $z=\tfrac{1-w}{t}$ you get: $$\int_0^{\infty}\frac{e^{-w/t}}{1-w}dw=e^{-1/t}\int_{-\infty}^{1/t}\frac{e^z}{z}dz=e^{-1/t}\mathrm{Ei}(1/t)$$

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