Consider again the same problem for an ideal gas of two- dimensional bosons confined to a surface of area A

Consider a system of N free bosons in d dimensions (in a hypercubic box of volume L^d). The temperature of the Bose- Einstein condensation comes from the expression

where C_d = 2π^d/2/Γ (d/2). With a suitable change of variables, we have

where

It is easy to see that the integral I diverges (I→ ∞) for d≤2, which means that there is no Bose-Einstein condensation below three dimensions.