Thin gravitational lens equation ($\mathbf{\beta }=\mathbf{\theta }-\mathbf{\alpha }$)

A thin lens maps apparent image angle $\mathbf{\theta }$ to unlensed source angle $\mathbf{\beta }$. In the usual outward reduced-deflection convention, $\mathbf{\alpha }={\nabla }_{\theta }\psi$ for a projected lensing potential. For an axisymmetric lens its magnitude is $4G{M}_{\mathrm{projected}}(<D_d\theta )D_{ds}/(c^2{D}_d{D}_s\theta )$. The mass here is projected inside a cylinder, not the three-dimensional mass inside a sphere.