\begin{table}%t3
%\centering
\caption{Theoretical formulae for the physical resolution ${\cal R}_Z$ in the vertical direction as functions of the number $M$ of source points,\protect $s$-parameter and disc semi-thickness $h$. The last column corresponds to a $3$-point, finite-difference method.}\label{tab:resolutionz}
\begin{tabular}{lcccc}\hline\hline
       &                   & & & Finite-diff. \\
       &                   &  \multicolumn{2}{c}{This method} & methods \\
Aspect &              &  &  $z  \rightarrow \pm h$  \\
ratio  &  $Z$           & $z \rightarrow Z$  & $(z ~ \ne Z)$ &  \\ \hline\noalign{\smallskip} 
  & $\ne 0$ & $\frac{1\pm\frac{Z}{h}}{2s(M-1)^{1/s}}$ & $\frac{1\pm\frac{Z}{h}}{2s(M-1)}$ & \\
$\frac{h}{R} \ge 1$ &&&& $\frac{1}{M-1}$\\
                      & $0$ & $\frac{1}{s}\left(\frac{2^{1-s}}{M-1}\right)^{1/s}$ &  $\frac{1}{s(M-1)}$ \\[0.3cm] 
$\left(\frac{h}{R}\right)^2 \ll 1$ & any & $\frac{1}{s}\left(\frac{2^{1-s}}{M-1}\right)^{1/s}$ & $\frac{1}{s(M-1)}$ & $\frac{1}{M-1}$ \\[0.3cm]
\underline{examples} &&\\
$(M,s)$\\
  $(8,\frac{1}{2})$ & & $0.08$ & $0.28$ & $0.14$\\
 $(32,\frac{1}{2})$ & & $4 \times 10^{-3}$ & $0.06$ & $0.03$ \\
 $(128,\frac{1}{2})$ & & $2 \times 10^{-4}$ & $0.02$ & $8 \times 10^{-3}$ \\
 $(128,\frac{1}{4})$ & & $10^{-7}$ & $0.03$ & $8 \times 10^{-3}$     \\
  \hline 
\end{tabular}
\end{table}