Multiple-image caustic of a softened isothermal lensing potential (${\beta }_c={\theta }_c[({\theta }_E/{\theta }_c{)}^{2/3}-1{]}^{3/2}$)

For $0<{\theta }_c<{\theta }_E$, the signed circular lens mapping $\beta =\theta [1-{\theta }_E/\sqrt{{\theta }^2+{\theta }_c^2}]$ has radial stationary points at ${\theta }_r^2=({\theta }_E{\theta }_c^2{)}^{2/3}-{\theta }_c^2$. Substituting them gives the displayed caustic radius. An offset point-like astronomical source has three images when $0<|\beta |<{\beta }_c$, a merging pair plus a separate outer image on the fold, and one image outside. At alignment, the two off-centre images form a ring while a central image remains. For ${\theta }_c\ge {\theta }_E$ the mapping is monotone and there is no finite multiple-image region. The zero-core singular limit loses the central image and has a different image-count exception.