content_image/377346.jpg content_image/377353.jpg content_image/377354.jpg content_image/377347.jpg content_image/377348.jpg content_image/377352.jpg content_image/377349.jpg content_image/377350.jpg content_image/377351.jpg
the term ˆ E is not present at all. The diaphanous wedge is analogous to the system of an annular region between two perfectly conducting cylinders, intercut by two semi- transparent, radially directed interfaces [36, 37]. Notably, while the energy expressions for a perfectly conducting wedge or circularly cylindrical shell require some regular- ization scheme in order to give numerical meaning, the energy expression obtained for the diaphanous wedge is immediately finite. A review of the Casimir wedge prob- lem and an early exposition of the issue we elaborate herein are found in Ref. [38]. 

Naturally, for the geometry of a perfectly conducting cylinder there is no divergent term ˆ E since there are no sharp corners. It turns out (c.f. the discussion in Section III of Ref. [34]) that the Casimir energy of a perfectly conducting cylindrical shell is 

Thus, all of the calculations in the following sections, which are carried out for general p , are valid also for a cylindrical shell by letting p → 1 and multiplying by an overall factor of 2. 

In the following we derive an analytical expression for the Casimir energy of a perfectly conducting wedge (mod- ulo a singular term as encountered in the past) and a perfectly conducting cylindrical shell, valid for arbitrary opening angles and all temperatures. This extends the calculations for the perfectly conducting wedge presented in [34], and simultaneously those for a circularly cylindri- cal conducting shell [8, 10], to the case of finite temper- ature. We show how the energy expression, which for T > 0 is the Helmholtz free energy, may be regularized by a scheme of Epstein-zeta functions to obtain a numer- ically useful expression. We show explicitly that the ex- pression thus obtained reduces to the previously derived zero temperature limit, and that the two leading terms of the high temperature asymptotic expansion, derived by Bordag, Nesterenko and Pirozhenko [21], are reproduced exactly as a special case. 

## II. CASIMIR-HELMHOLTZ FREE ENERGY OF WEDGE AND CYLINDER 

We take as our starting point the zero temperature energy derived for the geometry of a perfectly conducting wedge of opening angle α closed by a perfectly conducting cylindrical arc of radius a , derived in [34], shown on the left side of Fig. 1. 

Henceforth we shall focus on the term ˜ E $_{0}$, which may be written [34] 

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with 

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where we define the shorthand x 2 = a $^{2}$( k $^{2}$+ n $^{2}$ζ$^{2}$) where n is the index of refraction of the medium inside the wedge. We assume n to be constant with respect to ζ and uniform in space. Here ω = iζ is the reciprocal of imaginary (Euclidian) time. By means of partial inte- gration with respect to ζ , adding a trivial constant and noting that the integrand is symmetrical under ζ → − ζ and k → − k , this may be written on the familiar form ∞ ∞ 

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wherein we use the shorthand 

The Helmholtz free energy at T > 0 is obtained from the ‘trace-log’ formula (2.3) by compactifying the Eu- clidean time axis as is well known. Technically this amounts to the transition 

where ζ$_{j}$ = 2 πjT are the Matsubara frequencies. Chang- ing the integration variable from axial momentum k to x , the resulting expression for the finite part of the free energy may be written 

where 

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