Axial potential of a uniform hemispherical shell ($\varphi (h)=-\frac{2\pi G\rho }{3h}\left[b^3-a^3+\frac{3}{2}h(b^2-a^2)-(b^2+h^2{)}^{3/2}+(a^2+h^2{)}^{3/2}\right]$)

For constant mass density $\rho$ on the lower half of the spherical shell $a\le |{\mathbf{r}}^{\prime }|\le b$, with $0<a<b$, the displayed Newtonian gravitational potential applies at $(0,0,h)$ for every $h>0$, with zero at infinity. Use source spherical coordinates with polar angle $\alpha \in [\pi /2,\pi ]$. For a fixed source radius $s$,Multiplication by $-2\pi G\rho s^2$ and integration from $a$ to $b$ leaves the antiderivative $s^3/3+h{s}^2/2-(s^2+h^2{)}^{3/2}/3$, proving the formula. The entire positive axis is outside the matter, even for $a<h<b$, so there are no interior/exterior radial cases. The limit at the origin is $-\pi G\rho (b^2-a^2)$; far away the leading term is $-GM/h$, with total mass $M=2\pi \rho (b^3-a^3)/3$.