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I got to this integral, while proving some theorem in statistics:

$$\int_0^\infty \frac{e^{-t-\frac{x}{t}}}{t} \mathop{dt}$$

I have trouble evaluating it. I tried partial integration, tried substitution with some polynomial and some trigonometric functions. None of them helped, and Wolfram can't compute it either. Do you have a hint on how to solve this?

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    $\begingroup$ Perhaps that the beginning of chapter XVII (p.$355$) of Whittaker & Watson 'A Course Of Modern Analysis' will interest you (from $1927$ but still hard to replace !). The Bessel $J$ function is introduced there with just a little more general integral. $\endgroup$ Commented May 27, 2013 at 22:57

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With a little help from Maple, the integral is

$$\int_0^\infty \frac{e^{-t-\frac{x}{t}}}{t}\,dt = 2K_0(2\sqrt{x}),$$

where $K_0$ is the modified Bessel function of the second kind of order zero.

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  • $\begingroup$ ouch... So I guess that means I shouldn't be looking for a closed form expression. Right? $\endgroup$
    – Untitled
    Commented May 27, 2013 at 7:04
  • $\begingroup$ @Untitled There are people who know more about Bessel functions than I do, but I suspect that's as good a closed form as you will get. $\endgroup$
    – mrf
    Commented May 27, 2013 at 7:05
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    $\begingroup$ +1: DLMF 10.32.10 may be used as a reference for this integral. $\endgroup$ Commented May 27, 2013 at 7:39
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It worked on WolframAlpha after I hit the "Extended computation time" button:

enter image description here

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