Knowlpedia

Definition

Logarithmic radial derivative

The dilation derivative r partial_r, equal to differentiation with respect to log r.

For r>0r>0, the logarithmic radial derivative is D=r∂rD=r\partial_r. Setting s=log⁡rs=\log r gives

ddsf(es)=(Df)(es).\frac{d}{ds}f(e^s)=(Df)(e^s).

The proves this identity. It measures variation under multiplicative changes of radius, and D(ra)=araD(r^a)=a r^a.

Powers of the operator

The product rule gives D2f=r2f′′+rf′D^2f=r^2f''+rf'. Consequently

∂r2+r−1∂r=r−2D2,∂r2+r−1∂r−r−2=r−2(D2−1).\partial_r^2+r^{-1}\partial_r=r^{-2}D^2, \qquad \partial_r^2+r^{-1}\partial_r-r^{-2}=r^{-2}(D^2-1).

Multiplication by powers does not commute with DD: D(raf)=ra(D+a)fD(r^af)=r^a(D+a)f. These identities are useful for radial equations, but hold on r>0r>0; extension to the axis is a separate regularity question.