# 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: $$\frac{1}{C+i(A+B 0_+)}$$ where $$C$$, $$A$$, and $$B$$ are real numbers.

I am aware that the following is true: $$\operatorname{Im}\biggl(\frac{1}{\Delta\pm i0_+}\biggr)=P\biggl(\frac{1}{\Delta}\biggr)\mp i\pi\delta(\Delta)$$ however I am not sure how to apply that when there are more imaginary terms in the denominator... I would really appreciate some help!

• Welcome to the site! This is a good question, but do you happen to know of any resource you could link to that describes the meaning of $0_+$ in this context? Or if you could identify the paper where you found this notation, that might help too. Personally, I'm only familiar with this sort of notation from elementary particle physics, but I don't know if it's used the same way in thermal field theory. – David Z Sep 24 '18 at 7:20
• The paper is found at arxiv.org/abs/1506.06752 and the particular calculation at page 10! – alice_94 Sep 24 '18 at 7:25
• It is also used extensively in these notes arxiv.org/abs/1701.01554 (ex page 123)! – alice_94 Sep 24 '18 at 7:27