I am wondering if anyone knows of an integral representation for $\exp(-1/\sqrt{x})$ of the type $$ \exp(-1/\sqrt{x})=\int_\mathcal{I}d\xi\ f(\xi)\ e^{-\xi x}\ . $$ What would $\mathcal{I}$ and $f$ be? I am trying to integrate a complicated function, and this trick would bring the $x$ upstairs, facilitating my job (at the expense of an extra integration). Many thanks for your help.
Integral representation for $\exp(-1/\sqrt{x})$
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integration
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0With $x\to\infty$ what happens? – 2017-02-28
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0@MyGlasses not sure I understand the question... – 2017-02-28
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1The idea behing @MyGlasses' question is that, when $x\to\infty$, the LHS goes to $1$ while the integral on the RHS would go to $0$. Unfortunately, this argument would hold if $I\subseteq(0,+\infty)$, not for every $I\subset(-\infty,+\infty)$, hence it does not suffice to conclude. (Or, I missed an aspect of the argument.) – 2017-02-28
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0Try looking in Gradshteyn and Ryzhik's table of integrals. – 2017-02-28
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0Sorry, There is a similar to LHS by Laplace perhaps be helpful. For $\dfrac{1}{s}e^\frac{1}{\sqrt{s}}$ with function $$f(t)=\frac{1}{\sqrt{\pi x}} \int_0^\infty\exp(-0.25t^2x)J_0(2\sqrt{t})dt$$ Ref: **Spiegel & Murray, Laplace Transforms, Schaum's Outline of Laplace Transforms, McGraw-Hill (1965), p. 250.** – 2017-02-28
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0@MyGlasses thanks for your effort. I am slightly confused, though, as your $f(t)$ cannot be a function of $t$, since $t$ is also the integration variable....also, I cannot get hold of the book you cited. Could you please elaborate more? Many thanks, much appreciated... – 2017-02-28
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0I think your function maybe is found with Laplace transforms but a little hard and related with Bessel function, it's better see the reference: http://book4you.org/s/?q=Spiegel+Murray+Laplace+Transforms&e=1&t=0 – 2017-02-28
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0@MyGlasses the link is deleted by the legal owner, thx anyway – 2017-02-28
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0Sorry again, use http://www.olumcamp.ir/Downloads/Spiegel.djvu – 2017-02-28