the term proportional to $1/\psi_{eq}'$ in Eq.\[4\] yields a singularity.

The singular behavior is regularized by allowing for a small but finite resistivity, which smooths out the current divergence within a narrow boundary layer centered around the resonant surface. As a consequence, we can identify two separated regions, where different simplifications of the model equations apply: an “outer” linear ideal region, where Eq.\[4\] holds, and an “inner” linear or nonlinear resistive boundary layer, the narrowness of which allows geometrical simplifications, although non-ideal physics must be retained. The solutions of the equations in the two regions must match in a so called overlapping region. In other words, the “outer” solution provides the boundary conditions for the “inner” solution.

To simplify the matching procedure, it is convenient to introduce the stability parameter, $\Delta'$: $$\label{5}
\Delta'=\frac{1}{\widetilde{\psi}_{r_s}}\lim_{\epsilon\rightarrow
0}\left(\left.\frac{d \widetilde{\psi}}{d r}
\right|_{r=r_{s}+\epsilon} - \left.\frac{d \widetilde{\psi}}{d
r}\right|_{r=r_{s}-\epsilon} \right),$$ where $\widetilde{\psi}_{r_s}$ is the value of the perturbed magnetic flux at the resonant surface. Clearly, $\Delta'$ is fully determined once the outer magnetic flux is found by solving Eq.\[4\], and is a function of $m$, of $J_{eq}'/\psi_{eq}'$ and of the boundary conditions imposed on $\widetilde{\psi}$. The stability parameter defines the stability of the linear tearing mode since, generally, 