Weighted radial moment

For a real exponent $e$ , the weighted radial moment of a scalar function $f$ is

M_e(f)=\int_0^\infty r^e f(r)\,dr,

provided $r^ef(r)$ is integrable. The radial variable uses the one-dimensional measure $dr$ ; any geometric volume factor is included explicitly in the weight.

Parameters and constraints

If $f=f(r,z,t)$ , this moment is a function of $(z,t)$ . Requiring $M_e(f)=0$ means an identity at every parameter value, not just a finite collection of scalar equalities. Support in a fixed compact interval away from zero avoids endpoint issues for every real exponent. Weighted moments can also be defined componentwise for vector-valued functions.