Lippmann-Schwinger Equation

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Hi,
I have a less than acceptable grasp of Complex Analysis. I am confused about the derivation of the Lippmann-Schwinger equation. In the energy basis, a singularity is resolved (denominator: E-K) by adding a small complex term and performing integration over the positive reals. Can someone explain how this accomplishes the task of circumventing the singularity?
Thanks
 
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Roughly:
Look at where the denominator makes the expression undefined. It is no longer on the real axis, so integrating along the real axis is no longer problematic.
 

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