content_image/401408.jpg content_image/401413.jpg content_image/401414.jpg content_image/401419.jpg content_image/401409.jpg content_image/401415.jpg content_image/401416.jpg content_image/401417.jpg content_image/401420.jpg content_image/401421.jpg content_image/401427.jpg content_image/401424.jpg content_image/401423.jpg content_image/401418.jpg content_image/401422.jpg content_image/401426.jpg content_image/401425.jpg
Combining (4.4) and (4.10) we find, to leading order in τ , the expression in square brackets in (3.20): 

<img src='content_image/401408.jpg'>

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). 

## V. HIGH- τ ASYMPTOTICS: AGREEMENT WITH PREVIOUS RESULTS FOR CYLINDRICAL SHELL 

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 

<img src='content_image/401413.jpg'>

with 

<img src='content_image/401414.jpg'>

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): 

<img src='content_image/401419.jpg'>

where the correction term is exponential, wherewith 

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 ∞ 

<img src='content_image/401409.jpg'>

<img src='content_image/401415.jpg'>

<img src='content_image/401416.jpg'>

<img src='content_image/401417.jpg'>

<img src='content_image/401420.jpg'>

<img src='content_image/401421.jpg'>

With a litte calculation one ascertains that 

<img src='content_image/401427.jpg'>

When β → 0 it is simple to see from 

that 

<img src='content_image/401424.jpg'>

Hence we can safely ignore the term involving K at high τ . 

Combining this, the high- τ behavior of e ( τ, p ) is 

<img src='content_image/401423.jpg'>

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 

<img src='content_image/401418.jpg'>

As previously mentioned, ˜ E cyl = 2 ˜ E p $_{=1}$. With the ex- pansion (5.8) we find, using (5.3a), 

<img src='content_image/401422.jpg'>

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. 

<img src='content_image/401426.jpg'>

<img src='content_image/401425.jpg'>

