By Emilio Elizalde
Zeta-function regularization is a robust process in perturbation conception, and this ebook is a entire advisor for the coed of this topic. every thing is defined intimately, particularly the mathematical problems and tough issues, and a number of other functions are given to teach how the process works in perform, for instance within the Casimir impact, gravity and string conception, high-temperature section transition, topological symmetry breaking, and non-commutative spacetime. The formulae, a few of that are new, might be at once utilized in developing bodily significant, exact numerical calculations. The ebook acts either as a easy creation and a set of workouts in the event you are looking to observe this regularization strategy in perform. Thoroughly revised, up to date and increased, this new version contains novel, particular formulation at the basic quadratic, the Chowla-Selberg sequence case, an interaction with the Hadamard calculus, and likewise includes a clean bankruptcy on fresh cosmological purposes, together with the calculation of the vacuum power fluctuations at huge scale in braneworld and different types.
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Additional info for Ten Physical Applications of Spectral Zeta Functions
AN ; α1 , . . , αN ; c1 , . . 78) and for the generalized Epstein-like case: c EN (s; a; c) ≡ MNc (s; a1 , . . , aN ; 2, . . , 2; c1 , . . , cN ) ∞ = a1 (n1 + c1 )2 + · · · + aN (nN + cN )2 + c −s . ,nN =0 Consider the case of M2c . We need the result of the regularization theorem as applied to the double series ∞ 1 Sα (t, s) = n=1 ns+1 ∞ k=0 (−t)k αk n , k! 80) which converges for Re(s) > 0 large enough. 81) 42 2 Mathematical Formulas Involving the Different Zeta Functions where the contour C consists of the straight line Re(k) = k0 , with k0 fixed, 0 < k0 < 1, and of the semicircumference at infinity on the left of this line (see Fig.
1 + m; a2 , . . , aN ; c2 , . . , cN ) π (s − 1/2) c EN −1 (s − 1/2; a2 , . . , aN ; c2 , . . 93) j =2 where Kν is the modified Bessel function of the second kind. The recurrence starts with E1c (s; a1 ; c1 ) c−s ∼ (s) + ∞ m=0 c1/2−s 2 ∞ π a1 s−1/2 · m (−1)m (s + m) a1 m! 94) . Then E2c (s; a1 , a2 ; c1 , c2 ) a2−s (s) ∼ ∞ m=0 (−1)m (s + m) a1 m! 95) and so on. Expressions for the special case c = 0 are given in Ref.  (see also the preceding section and equations below). The very particular case, a1 = · · · = aN = 1, c1 = · · · = cN = 1 and α1 = · · · = αN = 2, simplifies considerably.
48) m=1 Finally, after correctly making the last step in the above proof, we end up with (α) SAB ∞ = a=0 (α) 1 (−1)a ζ (s + 1 − αa) + a! α ∞ SAB = a=0 a=s/α − − s (−1)a ζ (s + 1 − αa) + (−1) α a! 50) (α) ¯ where AB is the contribution of the curved part K of the contour C—which consists now of the line Re a = a0 , for fixed a0 such that 0 < a0 < 1 and by the semicircumference at infinity on the left (see Fig. 2) (α) AB ≡ K da ζ (s + 1 + αa) (a). 51) This contribution is non-zero for any value of s.
Ten Physical Applications of Spectral Zeta Functions by Emilio Elizalde