  The objective of this paper is the theoretical description of the
Mott-insulator to superfluid quantum phase transition of a Bose gas in an
optical lattice. In former works the Rayleigh-Schr\"odinger perturbation theory
was used within a mean-field approach, which yields partially non-physical
results since the degeneracy between two adjacent Mott lobes is not taken into
account. In order to correct such non-physical results we apply the
Brillouin-Wigner perturbation theory to the mean-field approximation of the
Bose-Hubbard model. Detailed explanations of how to use the Brillouin-Wigner
theory are presented, including a graphical approach that allows to efficiently
keep track of the respective analytic terms. To prove the validity of this
computation, the results are compared with other works. Besides the analytic
calculation of the phase boundary from Mott-insulator to superfluid phase, the
condensate density is also determined by simultaneously solving two algebraic
equations. The analytical and numerical results turn out to be physically
meaningful and can cover a region of system parameters inaccessible until now.
Our results are of particular interest provided an harmonic trap is added to
the former calculations in an homogeneous system, in view of describing an
experiment within the local density approximation. Thus, the paper represents
an essential preparatory work for determining the experimentally observed
wedding-cake structure of particle-density profile at both finite temperature
and hopping.
