How to understand Matsubara frequency sum?

[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!

 

[BigJDubs]

The two expressions are related, but not equal. The sum \sum_{n}\frac{\xi}{2\pi ni-\xi} is a regularized version of the other sum, which means that it has a regulator that is used to ensure that the sum is convergent and can be evaluated. The regularized version of the sum is the imaginary part of the sum and is equal to n_B(\xi). On the other hand, the unregularized version of the sum is equal to \pi \coth(\pi \xi). The key difference between the two is that the unregularized sum is divergent, while the regularized sum is convergent.