resonance, which implies that the principal value notation can be dropped. Note also that the integration limits are reduced to $x=\pm\pi/2$, which is halfway between the edge and the central resonance but still asymptotically far from the resistive layer (supposed to be infinitesimally small). We restrict our attention to tearing modes close to marginal stability, for which the constant-$\psi$ approximation [@FKR] is valid, i.e. $\widetilde{\psi}'_0 \ll
\widetilde{\psi}_0$ in the vicinity of the resonant surface. As a consequence, for these modes we can assume that $\widetilde{\psi}_0(x)/\widetilde{\psi}_{r_s} \cong 1$. Therefore, the integral $\int^{\pi/2}_{-\pi/2}dx[\sin(Kx)/\sin(x)]$ can be calculated for different values of $K$. For $K\geq 5$ its value is roughly constant and can be approximated with $\pi$ (the maximum error in this range is around $10\%$, and it reduces for large $K$). We remark that the matching theory described here is valid if $1/L \ll K \leq 1/\lambda$, where $L$ is a macroscopic length scale and $\lambda$ is the resistive boundary layer width. Indeed, if $K\lambda$ is too large the scale of the corrugation is of the same order of the boundary layer, thus making impossible to employ a perturbative technique, while if it is too small the effect of the corrugations is evanescent.

