seen that Ec is a function of volume only. This integral expression for the grand potential evaluates to: where Lis(x) is the polylogarithm function. The problem with this continuum approximation for a Bose gas is that the ground state has been effectively ignored, giving a degeneracy of zero for zero energy. This inaccuracy becomes serious when dealing with the Bose–Einstein condensate and will be dealt with in the next sections. As will be seen, even at low temperatures the above result is still useful for accurately describing the thermodynamics of just the un-condensed portion of the gas. Limit on number of particles in uncondensed phase, critical temperature The total number of particles is found from the grand potential by This increases monotonically with z (up to the maximum z = +1). The behaviour when approaching z = 1 is however crucially dependent on the value of α (i.e., dependent on whether the gas is 1D, 2D, 3D, whether it is in a flat or harmonic potential well). For α > 1, the number of particles only increases up to a finite maximum value, i.e., is finite at z = 1: where ζ(α) is the Riemann zeta function (using Liα(1) = ζ(α)). Thus, for a fixed number of particles , the largest possible value that β can have is a critical value βc. This corresponds to a critical temperature Tc=1/kBβc, below which the Thomas–Fermi approximation breaks down (the continuum of states simply can no longer support this many particles, at lower temperatures). The above equation can be solved for the critical temperature: For example, for the three-dimensional Bose gas in a box ( and using the above noted value of ) we get: For α ≤ 1, there is no upper limit on the number of particles ( diverges as z approaches 1), and thus for example for a gas in a one- or two-dimensional box ( and respectively) there is no critical temperature. Inclusion of the ground state The above problem raises the question for α > 1: if a Bose gas with a fixed number of particles is lowered down below the critical temperature, what happens? The problem here is that the Thomas–Fermi approximation has set the degeneracy of the ground state to zero, which is wrong. There is no ground state to accept the condensate and so particles simply 'disappear' from the continuum of states. It turns out,