= Multiple-image caustic of a softened isothermal lensing potential
{title2=$\beta_c=\theta_c[(\theta_E/\theta_c)^{2/3}-1]^{3/2}$}
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 $\theta_c\geq\theta_E$ 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.
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