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:

$$x^{-\alpha}$$