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If f(z) is defined to be the integration of rho(x) * (z-x)^(-1) from -infinity to +infinity. rho is in the following form

rho(x)=gamma * ( gamma^2+x^2 )^(-1).

Evaluate f(z) explicitly for Im(z)>0 and find its analytic continuation to Im(z)<0

First, I assumed gamma is real and positive and used the residue theorem, then I got

f(z)=pi * (z+i*gamma)^(-1), My question is how to find its analytic continuation to Im(z)<0.

Another question is that I found f(z) takes a different form if I assumed gamma is pure imaginary. It doesn't make sense to me.

Could you help me with these two questions? Thanks a lot.