Power law - Knowlpedia

A power law has the form $f(s)=Cs^a$ , where $s>0$ , $C\ne0$ , and $a\in\mathbb R$ is fixed; $s^a$ is the real power. It satisfies $f(\lambda s)=\lambda^a f(s)$ for $\lambda>0$ .

Asymptotic usage

An asymptotic power law $f(s)\sim Cs^a$ uses ratio convergence. The weaker estimate $|f(s)|\asymp s^a$ specifies an order up to constants. As $s\downarrow0$ , positive exponents decay and negative exponents grow. Naming the scale matters: if $s=\varepsilon^h$ , then $s^a=\varepsilon^{ha}$ .