I have this integral where it is singular at the point $p=0$, where I want to evaluate it on the solid contour given in the image below. My final goal is to find the integral in the range $(0,infty)$. My strategy is to use the solid contour and close the contour using the dashed lines then since there are no poles inside the closed contour, the integral in the closed contour is zero. Since the vertical dashed line contribution to the integral vanishes as $p=R rightarrow infty$, the integral in the range $(0,infty)$ is just negative times the rest of the contour (except the bottom solid contour which is what I want).
My problem is I don’t know how to find the integral of the upper dashed line plus the vertical solid line using my code. The small circular arc can be computed by hand using residues so that is no problem.
I know the integral along the whole solid line (except the circular arc) can be done using principal values but that does not allow me to find only the integral for the bottom solid line. Besides, I also don’t know how to find the principal value of the integral if one part of the domain is imaginary and the other part is real.
d = 2; func(p_) := 1/(Cosh(p/2)^(2/d) Tanh(p/2) Sqrt(1 - (Cosh(p/2)^(4 - 4/d) Tanh(p/2)^2)/(-0.419602)))