  In a previous work, we reported exact results of energies of the ground state
in the fractional quantum Hall effect (FQHE) regime for systems with up to
$N_{\text{e}} = 6$ electrons at the filling factor $\nu = 1/3$ by using the
method of complex polar coordinates. In this work, we display interesting
computational details of the previous calculation and extend the calculation to
$N_{\text{e}} = 7$ electrons at $\nu = 1/3$. Moreover, similar exact results
are derived at the filling $\nu = 1/5$ for systems with up to $N_{\text{e}} =
6$ electrons. The results that we obtained by analytical calculation are in
good agreement with their analogues ones derived by the method of Monte Carlo
in a precedent work.
