  The use of electrostatic ion beam traps require to set many potentials on the
electrodes (ten in our case), making the tuning much more difficult than with
quadrupole traps. In order to obtain the best trapping conditions, an
analytical formula giving the electrostatic potential inside the trap is
required. In this paper, we present a general method to calculate the
analytical expression of the electrostatic potential in any axisymmetric set of
electrodes. We use conformal mapping to simplify the geometry of the boundary.
The calculation is then performed in a space of simple geometry. We show that
this method, providing excellent accuracy, allows to obtain the potential on
the axis as an analytic function of the potentials applied to the electrodes,
thus leading to fast, accurate and efficient calculations. We conclude by
presenting stability maps depending on the potentials that enabled us to find
the good trapping conditions for oxygen 4+ at much higher energies than what
has been achieved until now.
