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:<math>\int_V e^{-\pi\langle \phi,S\phi\rangle}\, \mathcal D\phi</math>
where ''V'' is the function space and <math>\langle \cdot,\cdot\rangle</math> the [[L2 norm|L<sup>2</sup>]] inner product, and <math>\mathcal D\phi</math> the [[Wiener measure]]. The basic assumption on ''S'' is that it should be self-adjoint, and have discrete [[operator spectrum|spectrum]] λ<sub>1</sub>, λ<sub>2</sub>, λ<sub>3</sub>,
:<math> |\phi\rangle = \sum_i c_i |f_i\rangle \quad \text{with } c_i = \langle f_i | \phi \rangle. </math>
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This kind of Zeta-regularized functional determinant also appears when evaluating sums of the form {{nowrap|<math display="inline"> \sum_{n=0}^{\infty} \frac{1}{(n+a)} </math>}}. Integration over ''a'' gives <math display="inline"> \sum_{n=0}^{\infty}\ln(n+a) </math> which can just be considered as the logarithm of the determinant for a [[Harmonic oscillator]]. This last value is just equal to <math> -\partial _s \zeta_H(0,a) </math>, where <math> \zeta_H(s,a) </math> is the [[Hurwitz zeta function]].
==Practical example==
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== See also ==
* [[Abstract Wiener space]]
* [[Berezinian]]
* [[Fredholm determinant]]
* [[Fujikawa method]]
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==References==
*{{Citation | last1=Berline | first1=Nicole | last2=Getzler | first2=Ezra | last3=Vergne | first3=Michèle | title=Heat Kernels and Dirac Operators | isbn=978-3-540-20062-8 | year=2004| publisher=Springer }}
* {{Citation | last1=Branson | first1=Thomas P. | s2cid=14629173 | title=Q-curvature, spectral invariants, and representation theory | mr=2366932 | year=2007 | journal= Symmetry, Integrability and Geometry: Methods and Applications| issn=1815-0659 | volume=3 | pages=Paper 090, 31| arxiv=0709.2471 | doi=10.3842/SIGMA.2007.090 | bibcode=2007SIGMA...3..090B }}
* {{Citation | last1=Branson | first1=Thomas P. | title=The functional determinant | publisher=Seoul National University Research Institute of Mathematics Global Analysis Research Center | ___location=Seoul | series=Lecture Notes Series | mr=1325463 | year=1993 | volume=4}}
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