In many textbooks, the following fermionic Matsubara sum is given as a useful identity: $$T\sum_{\omega_n}\frac{1}{i\omega_n-\epsilon}=\frac{1}{e^{\epsilon/T}+1},$$ where $\omega_n=n\pi T$ for all odd $n$. However, when $\epsilon=0$, the LHS vanishes while the RHS equals 1/2. Did I misunderstand the identity?
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2I have no idea about this specific sum, but the general form here suggests that the l.h.s. either diverges for $\epsilon=0$ or doesn't represent what you actually want to compute. – ACuriousMind Feb 24 '16 at 12:07
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If one interprets the summation as the limit of symmetric partial sums (i.e. from $-N$ to $N$, then your left-hand side and right-hand side differs by $1/2$. I cannot (at a first sight) find the formula you have on that wiki page you refer to. Can you please be more precise in your reference? – mickep Feb 24 '16 at 13:41
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@mickep, it is in the first line of "Table of Matsubara Frequency Summations". I believe the sum is evaluated using contour integration. – anon Feb 25 '16 at 00:36