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for the case of continuously-varying <math>X_i</math>.
When ''H'' is an [[observable]], such as in quantum mechanics, it is common to express the summation as a [[trace (linear algebra)|trace]], so that
:<math>Z(\beta) = \mbox{tr}\left(\exp\left(-\beta H\right)\right)</math>
For the above notation to be valid, the observable must be [[trace class]], that is, of a form such that the summation exists and is bounded.
The number of variables <math>X_i</math> need not be [[countable]], in which case the sums are to be replaced by [[functional integral]]s. Although there are many notations for functional integrals, a common one would be
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