How to understand Matsubara frequency sum?

1. Sep 5, 2013

inempty

We know a standard matsubara frequency sum that $-\sum_{n}\frac{\xi}{2\pi ni-\xi}=n_B(\xi)$, but this looks contradictory to the well-known formula $\sum_{n}\frac{\xi}{n^2+\xi^2}=\pi \coth(\pi \xi)$ if we take the imaginary part of the former sum.

I know this matsubara frequency sum depends on the regulator since the sum itself diverges. But the imaginary part seems to be convergent well and the two results should conform. How to understand this contradiction?

Thank you!

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