Scaling of weighted radial moments

Statement

Let $L>0$ , let $A$ be a scalar, put $r=LR$ , and define $f_*(R)=Af(LR)$ . If the weighted integrals are absolutely convergent, then for real $e$ ,

\int_0^\infty R^e f_*(R)\,dR =AL^{-(e+1)}\int_0^\infty r^e f(r)\,dr.

This is the change-of-variables formula including the radial differential $dr=L\,dR$ .

Measure conventions

A cylindrical volume integral already contributes a factor $r$ ; its exponent must be included in $e$ . For example, $\int r f(r)\,dr$ acquires $L^{-2}$ , not $L^{-1}$ , when expressed in normalized variables with the above definition of $f_*$ . Parameters such as time are held fixed in this spatial substitution; differentiating them later also differentiates any parameter-dependent $A$ and $L$ .