  In two dimensions, the standard treatment of the scattering problem for a
delta-function potential, $v(\mathbf{r})=\mathfrak{z}\,\delta(\mathbf{r})$,
leads to a logarithmic singularity which is subsequently removed by a
renormalization of the coupling constant $\mathfrak{z}$. Recently, we have
developed a dynamical formulation of stationary scattering (DFSS) which offers
a singularity-free treatment of this potential. We elucidate the basic
mechanism responsible for the implicit regularization property of DFSS that
makes it avoid the logarithmic singularity one encounters in the standard
approach to this problem. We provide an alternative interpretation of this
singularity showing that it arises, because the standard treatment of the
problem takes into account contributions to the scattered wave whose momentum
is parallel to the detectors' screen. The renormalization schemes used for
removing this singularity has the effect of subtracting these unphysical
contributions, while DFSS has a built-in mechanics that achieves this goal.
