  We analyze the thermodynamic limit - modeled as the open-trap limit of an
isotropic harmonic potential - of an ideal, non-relativistic Bose gas with a
special emphasis on the phenomenon of Bose-Einstein condensation. This is
accomplished by the use of an asymptotic expansion of the grand potential,
which is derived by $\zeta$-regularization techniques. Herewith we can show,
that the singularity structure of this expansion is directly interwoven with
the phase structure of the system: In the non-condensation phase the expansion
has a form that resembles usual heat kernel expansions. By this, thermodynamic
observables are directly calculable. In contrast, the expansion exhibits a
singularity of infinite order above a critical density and a renormalization of
the chemical potential is needed to ensure well-defined thermodynamic
observables. Furthermore, the renormalization procedure forces the system to
exhibit condensation. In addition, we show that characteristic features of the
thermodynamic limit, like the critical density or the internal energy, are
entirely encoded in the coefficients of the asymptotic expansion.
