can then be expressed by $$\label{eq:phi_tot_reduction_factor}
\Phi_{\mathrm{tot}}(\lambda, x, z=0) = \Phi_0(\lambda, x, z=0) \; F(\lambda, z=0),$$ with the wavelength $\lambda \geq 0$ and $x = R - R_{\mathrm{0}}$ the position near a given radius $R_{\mathrm{0}}$. The reduction factor $F(\lambda)$ is generally between two limits: $\lim_{\lambda \to \infty} F(\lambda) = 1$ and $\lim_{\lambda \to 0} F(\lambda) = 0$. Therefore, for longer wavelengths $\lambda$, $\Phi_{\mathrm{tot}}$ (equation \[eq:phi\_tot\_reduction\_factor\]) behaves like the potential of a razor-thin disc and for shorter $\lambda$ the potential is reduced (Fig. \[fig:reductionfactor\]), e.g. @Kim:2002gl [@Wang:2010wr]. The transition between the regimes depends on the vertical density distribution.\
The vertical structure of a self-gravitating and isothermal sheet in hydrostatic equilibrium is described by a ${\, \mathrm{sech} }^{2}(z/z_{0})$ profile [@1942ApJ....95..329S], with the sound speed $c_{\mathrm{s}}$ and the scale-height $$\label{eq:scale-height_} 
z_{0} = \frac{c_{\mathrm{s}}^{2}}{\upi G \Sigma}.$$ The reduction factor is then given by (see Appendix \[App:Derivation of the reduction factor of the potential due to the disc thicknes\]) $$\label{eq:integralsechhyperbolfunction} 
F_{\mathrm{sech^2}}(\lambda, z_0)  =  \int_{-\infty}^{\infty}  \mathrm{e}^{- \frac{2 \upi}{\lambda} \mid h \mid} \; \frac{{\, \mathrm{sech} }^{2} \left( h /z_{0} \right)}   {2 z_{0}} \; dh.$$ This is basically the summation over all contributions to the total potential at the mid-plane, generated from infinitesimally thin layers at all vertical 