Is there an analytic expression for the following integral? \begin{equation} F = \int_a^\infty \dfrac{e^{-x}}{\sqrt{x-a}}\,dx \end{equation}
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1Make a substitution $x = y+a$. – Daniel Fischer Jul 14 '15 at 12:39
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But this change of variables doesn't seem to remove the singularity at $x=a$ (or $y=0$). Can you elaborate on your idea further? @DanielFischer – Maziar Noei Jul 14 '15 at 12:45
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But the singularity is integrable ($\frac{1}{\sqrt{y}}$), so we have no problem with that. The only remaining problem is to recognise the integral. – Daniel Fischer Jul 14 '15 at 12:48
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Given \begin{align} I = \int_{a}^{\infty} \frac{e^{-x}}{\sqrt{x-a}} \, dx \end{align} let $t = x-a$ to obtain \begin{align} I &= \int_{0}^{\infty} e^{-a - t} \, t^{(1/2) - 1} \, dt \\ &= e^{-a} \, \Gamma\left(\frac{1}{2}\right) = \sqrt{\pi} \, e^{-a} \end{align} where $\Gamma(x)$ is the gamma function.
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