What is the name of this theorem? (complex analysis)

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I am working on a problem to evaluate integrals with simple poles offset by ε above/below the real axis. So something like this

∫ [ f(x) / (x-x0-iε) ]

The answer is the sum of two integrals: the principal value of the integral with ε=0 plus the integral of iπδ(x-x0).

I have done the proof for the answer but my residue is off by a factor of a half (I have a factor of 2 in front of the delta, and I'm not sure why).

Does anyone know the name of this theorem and a place for the derivation?
 

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