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Given a power law equation: $$ y= a*b^x $$ Is it possible to approximate this equation with series of exponentials similar to the following? $$ y = c*\sum_i e^{k_i * x} $$ Thank you

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Yes, even a single exponential, because $a b^x$ is the same as $c e^{kx}$ with $a=c$ and $b=e^k$. But "power law" usually means a multiple of $x^{-r}$, not $b^x$. That can't be written as a sum of exponential decays, but can be written as a weighted average (= integral) of exponentials $e^{-xt}$ using the Gamma function: $$ x^{-r} = \frac1{\Gamma(r)} \int_0^\infty t^{r-1} e^{-xt} dt. $$

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  • $\begingroup$ Dear Noam, could you please take a look at mathoverflow.net/questions/252071/… ? I think my answer is correct, but I also think you would know for sure. $\endgroup$ – Will Jagy Oct 15 '16 at 15:56
  • $\begingroup$ Um, surely this request doesn't belong on this completely unrelated query... (and you have my e-mail address.) To your question, that seems right (though I didn't check the numbers), though probably unnecessary because splitting into several spinor genera requires discriminants of high valuation, and the Leech/Niemeier genus has dicsriminant whose valuations are as smaller as possible! But I've not studied this aspect of the theory carefully enough to be certain of this argument. $\endgroup$ – Noam D. Elkies Oct 15 '16 at 21:01
  • $\begingroup$ Noam, thank you. Between email and a comment of this type, I expected the comment to be less of an intrusion. In future I will use email or forget the thing. Gerry Myerson once made a similar comment when I called his attention to something in this way, so I guess I had this backwards. Sorry. $\endgroup$ – Will Jagy Oct 15 '16 at 21:25
  • $\begingroup$ @Noam - Could you suggest any textbooks/papers where the formula you quoted is derived? Would be of great help, thanks. $\endgroup$ – Moustache Feb 6 at 21:09

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