Power Laws

Representation o

Consider a power law with exponent \(\alpha > 0\):

\[x^{-\alpha}\]

This can be represented as an integral of exponentials:

\[x^{-\alpha} = \frac{1}{\Gamma(\alpha)} \int_0^{\infty} t^{\alpha-1} e^{-x t} dt\]

Proof: First, substitute the definition of the gamma function:

\[\frac{1}{\int_0^{\infty} b^{\alpha-1} e^{-b} db} \int_0^{\infty} t^{\alpha-1} e^{-x t} dt\]

Second, perform a variable substitution with \(xt = u \Rightarrow t = \frac{u}{x}, x dt = du\):

\[\frac{1}{\int_0^{\infty} b^{\alpha-1} e^{-b} db} \int_0^{\infty} \left( \frac{u}{x} \right)^{\alpha-1} e^{-u} \frac{1}{x} du\]

Third, simplifying:

\[\frac{x^{-\alpha}}{\int_0^{\infty} b^{\alpha-1} e^{-b} db} \int_0^{\infty} u^{\alpha-1} e^{-u} \frac{1}{x} du\]

Which yields:

\[x^{-\alpha}\]