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## VI. WEDGE WITH DIAPHANOUS ARC 

The above results can easily be extended to the case where the perfectly conducting arc is replaced by a di- aphanous arc, that is, a magnetodielectric interface so that the product n 2 = εµ is the same for radii both smaller than and greater than a . This geometry was considered at zero temperature in Ref. [34]. The electro- magnetic boundary conditions at the arc separate in a simple way in this case and the dependence on material properties enter only through the reflection coefficient 

The change in geometry leaves the energy expression (2.3) unaltered but for the simple replacement 

<img src='content_image/411617.jpg'>

This merely introduces a prefactor ξ 2 in all correction terms, and we can write down the result for the di- aphanous wedge, and simultaneously cylinder (by letting p = 1 and multiplying by 2 as discussed above), as: 

<img src='content_image/411618.jpg'>

<img src='content_image/411619.jpg'>

wherein 

<img src='content_image/411620.jpg'>

<img src='content_image/411621.jpg'>

<img src='content_image/411616.jpg'>

Since ξ enters the correction terms from renormaliza- tion only through the prefactor, generalization of the weak-coupling expansions (to leading order in ξ $^{2}$) con- sidered in [34] to nonzero τ is trivial. 

We have given for the first time results for the tem- perature dependence of the Casimir energy for a wedge, closed by a circular arc, all boundaries being perfectly conducting. This includes, as a special case, the perfectly conducting cylindrical shell case. (Except for that case, there is a divergent term, due to the corner where the circular arc meets the wedge boundaries, which we here simply omit.) The low temperature result agrees with the zero-temperature result found previously, except for what is probably an exponentially small correction, while the high temperature result agrees with that of Bordag, Nesterenko, and Pirozhenko for the case of a cylinder [21]. 

The work of KAM was supported in part by grants from the US National Science Foundation and the US Department of Energy. 

To evaluate a (possibly divergent) series Z = ∑ ∞ l =1 a$_{l}$ by Borel summation [41] we define the function 

If φ ( x ) is finite for sufficiently small x , we define the Borel transform as 

from which the Borel regularized value of the sum Z is Z = B (1). We consider the sum 

since B$_{1}$ = − 1 / 2 and B$_{3}$ = B$_{5}$ = B$_{7}$ = ... = 0. The Borel transform of the latter sum is thus 

## VII. CONCLUDING REMARKS 

## Appendix A: Evaluation of Eq. (4.9) by Borel summation 

<img src='content_image/411625.jpg'>

## Acknowledgments 

<img src='content_image/411622.jpg'>

<img src='content_image/411623.jpg'>

<img src='content_image/411624.jpg'>

