= Projected spherical Jeans residual
{title2=$p(R)=R^{-1}(R^2A(R))'$}
For a spherical nonrotating tracer with <luminosity density> $\lambda$, radial <velocity dispersion> $\sigma_r$, enclosed <luminosity> $L(r)$, and constant <mass-to-light ratio> $\Upsilon$, let $q=\lambda\sigma_r^2$ and $\gamma=G\Upsilon$. Define
$$
A(R)=\int_R^\infty\frac{q(r)r\,dr}{\sqrt{r^2-R^2}},\qquad p(R)=\mu(R)\sigma_p^2(R)-\gamma R^2\int_R^\infty\frac{\lambda(r)L(r)\,dr}{r^2\sqrt{r^2-R^2}}.
$$
The <Spherical Jeans equation> and the projected second <stellar velocity moment> give $p=2A+RA'$. To differentiate without a singular lower-end contribution, write $A(R)=\int_0^\infty q(\sqrt{R^2+z^2})\,dz$. If $R^2A$ vanishes at both endpoints, $\int_0^\infty Rp(R)\,dR=0$. A finite outer aperture instead retains its boundary stress term.
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