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$$ I(y;c,\lambda) \equiv\int_{0}^\infty \frac{\lambda c}{x} \exp\left(-\lambda x\right)\exp\left(-\frac{c}{x}y\right)dx$$ where $c,\lambda>0$.

Q: Can this integration be made in analytic form (series are acceptable, but closed form preferred)?

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Hint: $~\displaystyle\int_0^\infty\exp\bigg(-ax-\frac bx\bigg)~\frac{dx}x~=~2K_0\Big(2\sqrt{ab}\Big),~$ for positive values of a and b. See Bessel function for more information.

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