# Is there a way to express an power law decay as a series of exponentials? [closed]

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

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: $$x^{-r} = \frac1{\Gamma(r)} \int_0^\infty t^{r-1} e^{-xt} dt.$$