content_image/428409.jpg content_image/428412.jpg content_image/428418.jpg content_image/428408.jpg content_image/428414.jpg content_image/428411.jpg content_image/428413.jpg content_image/428417.jpg content_image/428415.jpg content_image/428416.jpg content_image/428410.jpg
The generating function of the Bernoulli numbers is 

to evaluate 

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where ψ ( n $^{)}$( x ) is the polygamma function, whose inte- gral representation was recognized (Ref. [39] Eq. 6.4.1) by making the substitution u = xt . Thus we evaluate the integral to 

<img src='content_image/428412.jpg'>

where from the requirement that B (0) = 0 we see that the integration constant is zero. Thus we find the Borel value of the sum (A3) to be 

noting that ψ $^{(2) = 1}$− γ . 

## Appendix B: Correspondence with high- T asymptotics for the cylinder in vacuum 

The heat kernel expansion for high temperatures cal- culated in Ref. [21] for the cylindrical shell in vacuum begins 

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[1] H. B. G. Casimir, Proc. Kon. Ned. Akad. Wetensch. 51 , 793 (1948). 

[2] S. K. Lamoreaux, Phys. Rev. Lett. 78 , 5 (1997). 

[3] M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Advances in the Casimir Effect (Oxford: Oxford University Press, 2009) 

[4] K. A. Milton, J. Phys. A: Math. Gen. 37 , R209 (2004). 

[5] K. A. Milton The Casimir Effect: Physical Manifesta- 

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where the ‘zeta determinant’ ζ $^{′}$(0) is a constant defined in Ref. [21] and we have inserted their value [20, 21] 

The term proportional to τ ln τ is obviously identical to our expression in Eq. (5.10). We consider only the term linear in τ . Comparison with (5.10) gives, with minimal manipulation, that the asymptotes correspond exactly according to ˜ E cyl = 2 ˜ E p $_{=1}$, provided 

<img src='content_image/428411.jpg'>

In Appendix B of [21] we find the following expression 

<img src='content_image/428413.jpg'>

with t = 1 / √ 1 + y $^{2}$. Let us call the two integrals in (B4) X$_{0}$ and X$_{m}$ , where the latter is the integral inside the sum. After a partial integration and, in the case of X$_{m}$ , a substitution ym = x , these can be written on the familiar form 

Comparing with (5.2) we see that 

<img src='content_image/428417.jpg'>

since 1 4 ∫ ∞ 0 dxx $^{4}$/ (1+ x $^{2}$)3 = 3 π/ 64. We have thus shown the correspondence analytically. 

tions of the Zero-Point Energy (World Scientific, Singa- pore, 2001). 

[6] E. M. Lifshitz, Zh. Eksp. Teor. Fiz. 29 , 94 (1955) [Sov. Phys. JETP 2 , 73 (1956)]. 

[7] T.H. Boyer, Phys. Rev. 174 , 1764 (1968). 

[8] L. L. DeRaad, Jr. and K. A. Milton, Ann. Phys. 186 , 229 (1981) 

[9] P. Gosdzinsky and A. Romeo, Phys. Lett. B 441 , 265 

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