zero temperature degenerate free fermion gas, where µ > 0 is the Fermi energy. In fact, the function Li$_{ν}$( z ) has a branch cut along the real axis from z = 1 to ∞ and the bosonic free energy is ill-defined at z = ∞ , in contrast with fermions. Using the analytic continuation of the asymptotic behavior from positive integer ν to half-integer values, we define the scaling functions b$_{d}$ as follows: − Li$_{ν}$( − z ) ≈ log $^{ν}$( z ) Γ( ν + 1) as z → ∞ (26) F = − mµ 2 4 π b$_{2}$ ( T/µ ) ( d = 2) (27) F = − $_{2}$√$_{2}$$_{m}$ 3 / $^{2}$µ5 / 2 15 π 2 b$_{3}$ ( T/µ ) ( d = 3) n = $_{2}$√$_{2}$ 15 π $_{2}$( mµ ) 3 / 2 ( 5 2 b$_{3}$ − T µ b ′ 3 ) (28) s = $_{2}$√$_{2}$ 15 π $_{2}$( mµ ) 3 / 2 b ′ $_{3}$( T/µ ) n = mµ 4 π ( 2 b$_{2}$ − T µ b ′ 2 ) (29) s = mµ 4 π b ′ 2 The above normalizations are defined such that b$_{d}$ (0) = 1 for a single free fermion. One can verify that the above d = 3 expression is the standard result for a zero temperature, single component, degenerate fermion gas where µ is the Fermi energy. Other thermodynamic quantities follow from the free energy. The pressure p = −F . The density is n = − ∂ F /∂µ and the entropy density is s = − ∂ F /∂T . Using the scaling form in eq. (27), one obtains for d = 3: where b ′ is the derivative with respect to its argument T/µ . (Henceforth, g ′ will always refer to the derivative of the function g with respect to it’s argument µ/T or T/µ as defined above.) The analogous formulas in 2 dimensions are: 13