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The first example in which zeta function regularization is available appears in the Casimir effect, which is in a flat space with the bulk contributions of the quantum field in three space dimensions. In this case we must calculate the value of Riemann zeta function at –3, which diverges explicitly. However, it can be [[analytic continuation|analytically continued]] to ''s'' = –3 where hopefully there is no pole, thus giving a finite value to the expression. A detailed example of this regularization at work is given in the article on the detail example of the [[Casimir effect#Derivation of Casimir effect assuming zeta-regularization|Casimir effect]], where the resulting sum is very explicitly the [[Riemann zeta function|Riemann zeta-function]] (and where the seemingly legerdemain analytic continuation removes an additive infinity, leaving a physically significant finite number).
An example of zeta-function regularization is the calculation of the [[vacuum expectation value]] of the [[energy]] of a particle field in [[quantum field theory]]. More generally, the zeta-function approach can be used to regularize the whole [[energy–momentum tensor]] both in flat and curved spacetime. {{ref|Mo97}} {{ref|BCEMZ}} <ref>{{Cite book |last=Fermi |first=D. |url=https://www.worldscientific.com/worldscibooks/10.1142/10570#t=aboutBook |title=Local zeta regularization and the scalar Casimir effect. A general approach based on integral kernels |last2=Pizzocchero |first2=L. |publisher=World Scientific Publishing |year=2017 |isbn=978-981-3224-99-5 |language=English |doi=10.1142/10570}}</ref>
The unregulated value of the energy is given by a summation over the [[zero-point energy]] of all of the excitation modes of the vacuum:
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* {{Citation | last1=Seeley | first1=R. T. | editor1-last=Calderón | editor1-first=Alberto P. | title=Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) | publisher=Amer. Math. Soc. | ___location=Providence, R.I. | series=Proceedings of Symposia in Pure Mathematics | isbn=978-0-8218-1410-9 | mr=0237943 | date=1967 | volume=10 | chapter=Complex powers of an elliptic operator | pages=288–307}}
* {{note|Do76}} {{Citation | last1=Dowker | first1=J. S. | last2=Critchley | first2=R. | title=Effective Lagrangian and energy–momentum tensor in de Sitter space | journal=Physical Review D | volume=13 | pages=3224–3232 | date=1976 | issue=12 | doi=10.1103/PhysRevD.13.3224| bibcode=1976PhRvD..13.3224D }}
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