Functional determinant: Difference between revisions

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Since the analytic continuation of the zeta function is regular at zero, this can be rigorously adopted as a definition of the determinant.

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>,}}. integrationIntegration over ''a'' gives <math display="inline"> \sum_{n=0}^{\infty}\ln(n+a) </math> which itcan just can be considered as the logarithm of the determinant for a [[Harmonic oscillator]]. thisThis 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==