I'm trying to find out how power law decays can be represented, or approximated, by exponential functions. Any papers or textbook suggestions would be particularly helpful. But in particular, on the following closed MathOverflow question
Is there a way to express an power law decay as a series of exponentials? [closed]
Noam D. Elkies suggested the following relationship for relating power law decay to the exponential function:
$x^{-r} = \frac{1}{\Gamma(r)} \int_{0}^{\infty} t^{r-1} e^{-xt} dt.$
I have looked for some time online, and also tried and failed to prove this result myself. I'm hoping someone might be able to either show why it's true, tell me which paper/textbook I can find it, or both of the above.
If there are other ways of representing/approximating power law decay in terms of exponentials I'd be interested in those too. At present this is the only paper I have been able to find: