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How can we calculate the imaginary part of a fraction that has a term $i0_+$ in the denominator?

I have recently started dealing with thermal field theory for fermions and I am faced with a paper that, at some point, tries to calculate the imaginary part of a fraction that looks like: $$\frac{1}{...
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Fermion boundary condition for a thermal compact circle

Is this true that for fermion statistical systems in the thermal phase, with Euclidean time, $$ \beta=1/T=t_E $$ the Euclidean time will be chosen to be anti-periodic for fermion boundary ...
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Matsubara sum for particle number

I am trying to derive the electron number equation for some thermal system. There is something wrong in my calculation. The action has the following form: $S = \sum_{k,n \in \rm{odd}} \bar{\psi}[-\...