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The zeta-regularization is useful as it can often be used in a way such that the various symmetries of the physical system are preserved. Besides the Casimir effect, zeta-function regularization is used in [[conformal field theory]] and in fixing the critical [[spacetime]] dimension of [[string theory]].
kaki
==Relation to other regularizations==
Zeta-function regularization gives a nice analytic structure to any sums over an [[arithmetic function]] <math>f(n)</math>. Such sums are known as [[Dirichlet series]]. The regularized form
:<math>\tilde{f}(s) = \sum_{n=1}^\infty f(n)n^{-s}</math>
converts divergences of the sum into [[simple pole]]s on the complex ''s''-plane. In numerical calculations, the zeta-function regularization is inappropriate, as it is extremely slow to converge. For numerical purposes, a more rapidly converging sum is the exponential regularization, given by
:<math>F(t)=\sum_{n=1}^\infty f(n) e^{-tn}</math>
This is sometimes called the [[Z-transform]] of ''f'', where ''z''=exp(-''t''). The analytic structure of the exponential and zeta-regularizations are related. By expanding the exponential sum as a [[Laurent series]]
:<math>F(t)=\frac{a_N}{t^N} + \frac{a_{N-1}}{t^{N-1}} + \ldots</math>
one finds that the zeta-series has the structure
:<math>\tilde{f}(s) = \frac{a_N}{s-N} + \ldots</math>
The structure of the exponential and zeta-regulators are related by means of the [[Mellin transform]]. The one may be converted to the other by making use of the integral representation of the [[Gamma function]]:
:<math>\Gamma(s+1)=\int_0^\infty x^s e^{-x}dx</math>
which lead to the identity
:<math>\Gamma(s+1) \tilde{f}(s+1) =
\int_0^\infty t^s F(t) dt</math>
relating the exponential and zeta-regulators, and converting poles in the s-plane to divergent terms in the Laurent series.
==Heat kernel regularization==
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