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

Question

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?

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. — Raymond Manzoni, May 27 '13 at 22:57

score 4 · Accepted Answer · answered May 27 '13 at 06:58

4

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.

answered May 27 '13 at 06:58

mrf

43,639

ouch... So I guess that means I shouldn't be looking for a closed form expression. Right? – Untitled May 27 '13 at 07:04
@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. – mrf May 27 '13 at 07:05
1

+1: DLMF 10.32.10 may be used as a reference for this integral. – Raymond Manzoni May 27 '13 at 07:39

score 1 · Answer 2 · answered May 27 '13 at 12:03

1

It worked on WolframAlpha after I hit the "Extended computation time" button:

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answered May 27 '13 at 12:03

Mark McClure

30,510

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

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