8 The generating function of the Bernoulli numbers is to evaluate 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 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 . The heat kernel expansion for high temperatures cal- culated in Ref. [21] for the cylindrical shell in vacuum begins Appendix B: Correspondence with high- T asymptotics for the cylinder in vacuum 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 In Appendix B of [21] we find the following expression 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 since 1 4 ∫ ∞ 0 dxx $^{4}$/ (1+ x $^{2}$)3 = 3 π/ 64. We have thus shown the correspondence analytically. ∞ ∑ l =1 B$_{l}$ l ! y l = y e y − 1 − 1 (A5) $^{d}$B dx ( x ) = 1 x ∫ ∞ 0 dte − t ∞ ∑ l =1 B$_{l}$ l ! ( xt ) l = ∫ ∞ 0 dt te − t e xt − 1 − 1 x = x − 2 ∫ ∞ 0 duue − u x +1 x $^{1}$− e − u − 1 x = x − $^{2}$ψ$^{(1)}$( x +1 x $^{)}$− 1 x , (A6) a$_{3}$$_{/}$$_{2}$ (4 π ) 3 / 2 = 3 64 a . (B2) πaζ $^{′}$(0) + 2 C (1) = π 64 [6 γ − 5 + 6 ln a 2 ] . (B3) B ( x ) + ln x = ∫ x dy y $_{2}$ψ$^{(1)}$( y +1 y ) = − ∫ x +1 x dv ψ $^{(1)}$( v ) $^{=}$− ψ ( x +1 x ) + constant , (A7) πaζ $^{′}$(0) = ∫ ∞ 0 dyy d dy $^{ln[1}$− y $^{2}$λ2$_{0}$( y )] + 2 ∞ ∑ m =1 m ∫ ∞ 0 dyy d dy { $^{ln[1}$− m $^{2}$y$^{2}$λ2 $_{m}$( my ) + y $^{4}$t6 4 m $_{2}$] } + π 32 ( 3 γ − 4 + 3 ln a 2 ) (B4) ∞ ∑ l =1 B$_{2}$$_{l}$ 2 l = 1 2 $^{+}$B(1) = γ − 1 $_{2}$, (A8) X$_{0}$ $^{=}$− ∫ ∞ 0 dx $^{ln[1}$− x $^{2}$λ2$_{0}$( x )]] (B5a) X$_{m}$ $^{=}$− 1 m ∫ ∞ 0 dx { $^{ln[1}$− x $^{2}$λ2 $_{m}$( x )] + z 4 4( m 2 + x $^{2}$)$_{3}$] } . (B5b) πaζ $^{′}$(0) + 2 C (1) = ∫ ∞ 0 dxx 4 4(1 + x $^{2}$)3 + π 32 ( 3 γ − 4 + 3 ln a 2 ) = π 64 [6 γ − 5 + 6 ln a 2 ] (B6) ˜ E cyl ∼ − T 2 ζ $^{′}$(0) − a$_{3}$$_{/}$$_{2}$ (4 π ) 3 / $_{2}$T ln T + . . . $^{=}$− τ 8 πa 2 [ 2 aζ $^{′}$(0) − 3 16 ln 2 πa + 3 16 ln τ + ... ] (B1) tions of the Zero-Point Energy (World Scientific, Singa- pore, 2001). [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- [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