In order to obtain a mathematically correct expression for the potential of the singular isothermal sphere some care is needed. It is the analogous of the potential on an infinite, rectilinear charge distribution. The reference point for the potential cannot be fixed at the origin, and it cannot be fixed at infinity. What can (and should!) be done is to compute the potential as follows

phi(r) - phi(r0) = -G Integral_r^r0 M(x) x^-2 dx,

where M(x) is the mass of the isothermal sphere contained in the sphere of radius x, centered at the origin. Of course, the value for phi(r0) is fully arbitrary, the usual choice is 0.

In order to obtain a mathematically correct expression for the potential of the singular isothermal sphere some care is needed. It is the analogous of the potential on an infinite, rectilinear charge distribution. The reference point for the potential cannot be fixed at the origin, and it cannot be fixed at infinity. What can (and should!) be done is to compute the potential as follows

$$\phi(r) - \phi(r_0) = -G\int_r^{r_0} M(x) x^{-2} \mathrm dx,$$

where $M(x)$ is the mass of the isothermal sphere contained in the sphere of radius $x$, centered at the origin. Of course, the value for $\phi(r_0)$ is fully arbitrary, the usual choice is $0$.