3 and where we have defined the dimensionless tempera- ture term subtracted, Similarly to the case at zero temperature this simple ex- pression is in need of regularization in order to give nu- merical meaning since it is formally divergent. We here follow a scheme closely reminiscent of that of Milton and DeRaad [8], and particularly Milton, Nesterenko and Nesterenko [10] (cf. also Appendix A of Ref. [34]). As follows from the uniform asymptotic expansion of modified cylindrical Bessel functions, e.g. § 9.7 of Ref. [39], the logarithmic factor in the integrand of (2.7) has the asymptotic behavior To see how this behavior gives rise to a formal diver- gence, consider the case of large m for which where we substituted y 2 = x 2 − j $^{2}$τ2 and defined the shorthand The three terms of (3.2b) correspond to the three terms of the integrand of (3.2a). All of the terms of (3.2b) clearly diverge when summed over j and m . The first step in regularization is to add and subtract the asymptotic behavior (3.1) in the form (3.2a) where we define the energy with the leading asymptotic III. REGULARIZATION OF THE FREE ENERGY EXPRESSION τ = 2 πnaT. (2.8) $^{ln[1}$− x $^{2}$λ2 $_{mp}$] ∼ − x 4 4( m $^{2}$p2 + x $^{2}$)$_{3}$, m, x → ∞ ; (3.1a) $^{ln[1}$− x $^{2}$λ2$_{0}$] ∼ − x 4 4(1 + x $^{2}$)$_{3}$, x → ∞ . (3.1b) e$_{m,j}$ ∼ − 1 4 ∫ ∞ jτ dx x 5 √ x 2 − j $^{2}$τ$^{2}$( m $^{2}$p2 + x $^{2}$)3 $^{=}$− 1 4 ∫ ∞ 0 dy ( y 4 + 2 j $^{2}$τ$^{2}$y2 + j $^{4}$τ$^{4}$) ( m $^{2}$p2 + j $^{2}$τ2 + y $^{2}$)3 (3.2a) $^{=}$− 3 π 64 ϕ − πj $^{2}$τ2 32 ϕ 3 − 3 πj $^{4}$τ4 64 ϕ 5 . (3.2b) ϕ = √ p $^{2}$m2 + τ $^{2}$j$^{2}$. (3.3) ˜ E = ¯ E $^{+}$E ∞ (3.4) ¯ E = T πa ∞ ∑ m =0$^{′}$ ∞ ∑ j =0$^{′}$ ˜ e$_{m,j}$ (3.5a) ˜ e$_{m,j}$ = ∫ ∞ jτ dx x √ x 2 − j $^{2}$τ2 × { $^{ln[1}$− x $^{2}$λ2 $_{mp}$( x )] + x 4 4( m $^{2}$p2 + x $^{2}$)3 } ; (3.5b) ˜ e$_{0}$$_{,j}$ = ∫ ∞ jτ dx x √ x 2 − j $^{2}$τ2 × { $^{ln[1}$− x $^{2}$λ2$_{0}$( x )] + x 4 4(1 + x $^{2}$)3 } , (3.5c) E ∞ = − T 4 πa ∞ ∑ m =0$^{′}$ ∞ ∑ j =0$^{′}$ ∫ ∞ 0 dy ( y 4 + 2 j $^{2}$τ$^{2}$y2 + j $^{4}$τ$^{4}$) ( ˜ ϕ 2 + y $^{2}$)3 . (3.6) ˜ ϕ = { √ 1 + τ $^{2}$j$^{2}$, m = 0 p $^{2}$m2 + τ $^{2}$j$^{2}$, m ≥ 1 . (3.7) E ∞ $^{=}$− T 4 πa lim s → 0 + ∞ ∑ m =0$^{′}$ ∞ ∑ j =0$^{′}$ ∫ ∞ 0 dyy − s ( ˜ ϕ 2 + y $^{2}$)3 × ( y 4 + 2 j $^{2}$τ$^{2}$y2 + j $^{4}$τ$^{4}$) ∼ − T 64 a lim s → 0 + ∞ ∑ m =0$^{′}$ ∞ ∑ j =0$^{′}$ { 3 ˜ ϕ 1+ s + 2 j $^{2}$τ2 ˜ ϕ 3+ s + 3 j $^{4}$τ4 ˜ ϕ 5+ s } , (3.8) ∫ ∞ 0 dyy 4 − s ( ˜ ϕ 2 + y $^{2}$)3 = π $^{(1}$− s )(3 − s ) 16 ˜ ϕ 1+ s sec πs 2 , (3.9) ∂$_{τ}$ 1 ˜ ϕ q $^{=}$− qτj 2 ˜ ϕ q $_{+2}$; (3.10a) ∂ 2 τ 1 ˜ ϕ q $^{=}$− qj 2 ˜ ϕ q +2 + q ( q + 2) τ $^{2}$j4 ˜ ϕ q +4 (3.10b) E ∞ $^{=}$− T 64 a $^{(3}$− 3 τ∂$_{τ}$ + τ $^{2}$∂2 $_{τ}$) × lim s → 0 + ∞ ∑ m =0$^{′}$ ∞ ∑ j =0$^{′}$ 1 ˜ ϕ 1+ $_{s}$. (3.11) to write which is valid for − 1 < s < 5, so it may be used for s near 0, near 2, or near 4. We use the relations [ ∂$_{τ}$ = ∂/∂τ ] where as in Eq. (3.2b) we have used the evaluation, $^{To regularize}$E ∞ we introduce the small quantity s and write wherein and the additional, non-regularized energy