6 Combining (4.4) and (4.10) we find, to leading order in τ , the expression in square brackets in (3.20): With a litte calculation one ascertains that and using we regain exactly the zero temperature result (4.1). As illustrated in Fig. 3 this limit is reached very rapidly as τ → 0. While we have ascertained in the above that the correction term in (4.11) is at least of order τ $^{4}$, there is reason to suspect that the behavior is in fact exponential, as is the case for S as seen from Eq. (4.3). We will finally determine the asymptotic behavior in the limit τ ≫ 1. Here the contribution from ¯ E is given by the zeroth Matsubara term only. Consider the reduced energy e ( τ, p ) of Eq. (3.20) in which with Some numerical values are These values were obtained with Mathematica, including 100 terms in the sum while checking convergence. The high- τ behavior of S is given immediately by Eq. (3.18): where the correction term is exponential, wherewith The high temperature asymptotics of perfectly con- ducting spherical and cylindrical shells with vacuum in- side and outside were calculated by Bordag, Nesterenko and Pirozhenko [20, 21] using the method of heat kernel coefficients. They, like us, found that the two leading order terms were of order T and T ln T as T → ∞ . The latter of these terms had been worked out some time previously by Balian and Duplantier[19], who also found an approximate (though not very accurate) value for the former. The result of the calculations reported in [21] was, in our notation As previously mentioned, ˜ E cyl = 2 ˜ E p $_{=1}$. With the ex- pansion (5.8) we find, using (5.3a), Hence we can safely ignore the term involving K at high τ . Combining this, the high- τ behavior of e ( τ, p ) is The slight numerical difference we believe to be due to the approximate numerical method used in [21]. We show analytically in Appendix B that the correspondence is in fact exact. To study the behavior of K ( τ ) and its derivatives it is useful to define β = 1 /τ and write K ( τ ) = $^{β}$H ( β ) with $^{[1 + 2}$K $^{+}$S] ∼ − 2 τ ln(2 π/p $^{) +}$O( τ $^{4}$) , τ → 0 (4.11) $^{(3}$− 3 τ ∂$_{τ}$ + τ $^{2}$∂2 $_{τ}$) 1 τ = 8 τ (4.12) When β → 0 it is simple to see from that V. HIGH- τ ASYMPTOTICS: AGREEMENT WITH PREVIOUS RESULTS FOR CYLINDRICAL SHELL 4 τ π ∞ ∑ m =0$^{′}$ ∞ ∑ j =0$^{′}$ ˜ e$_{m,j}$ ( τ, p ) ∼ 2 τ π C ( p ) , τ → ∞ , (5.1) C ( p ) = 1 2 ∫ ∞ 0 dx [ $^{ln(1}$− x $^{2}$λ2$_{0}$) + x 4 4(1 + x $^{2}$)3 ] + ∞ ∑ m $_{=1}$∫ ∞ 0 dx [ ln(1 − x $^{2}$λ2 $_{mp}$) + x 4 4( m $^{2}$p2 + x $^{2}$)3 ] . (5.2) C $^{(1) =}$− 0 . 75814; (5.3a) C $^{(2) =}$− 0 . 76558; (5.3b) C $^{(3) =}$− 0 . 76645 . (5.3c) ˜ E cyl ∼ − 0 . 22924 T a − 3 T 64 a ln aT 2 $^{+}$O( T − $^{1}$) $^{=}$− τ 8 πa 2 [ 0 . 44237 + 3 16 ln τ $^{+}$O( τ − $^{2}$) ] . (5.9) S ( τ, p ) ∼ 2 p ( γ − ln 4 πp + ln τ ) , τ → ∞ (5.4) $^{τ ∂$_{τ}$}$S ∼ 2 p ; τ $^{2}$∂2 τ S ∼ − 2 p . H ( β ) = ∞ ∑ j =1 j − √ j 2 + β 2 j j 2 + β 2 . (5.5) 2 ˜ E p =1 ∼ − τ 8 πa 2 [ 6 γ − $^{5}$− 6 ln 4 π 32 − 4 π C (1) + 3 16 ln τ + ... ] $^{=}$− τ 8 πa 2 [ 0 . 44270 + 3 16 ln τ + ... ] . (5.10) e ( τ, p ) ∼ τ [ 2 C ( p ) π − 3 p + 6( γ − ln 4 πp $^{)}$− 8 64 p − 3 32 p ln τ ] . (5.8) H ( β ) ∼ 1 $^{$_{2}$β}$H $^{′}$( β ) ∼ 1 $_{2}$β $^{2}$H $^{′′}$( β ) ∼ − 1 $_{2}$β$^{2}$ζ (3) , β → 0 . H $^{′′}$( β ) = ∞ ∑ j =1 2 β 2 − j 2 ( j 2 + β $^{2}$) 5 2 ; (5.7b) H $^{′}$( β ) = − β ∞ ∑ j =1 ( j 2 + β $^{2}$) − 3 2 ; (5.7a) τ $^{(3}$−3 τ ∂$_{τ}$ + τ $^{2}$∂2 $^{$_{τ}$)}$K ( τ ) = (8+7 β∂$_{β}$ + β $^{2}$∂2 $^{$_{β}$)}$H ( β ) . (5.6)