I can use the triangular contour shown below to conclude:

via Cauchy's Theorem and let the diameter of the center indentation go to zero. We know that for a simple pole, when the diameter goes to zero, the integral over the indentation is where is the angular length. For the contour below and in the positive sense, then:

I'll do the top line:

You should get for the vertical line. Add them up, then the integral over these lines is . But over the indented contour was which must mean the Principal Value over the diagonal line is zero, that is:

which you can show by letting , taking antiderivatives, and taking the limit:

In all cases, I chose a branch of analytic over the entire contour:

You may wish to try this using a lower-triangular contour and using a suitable determination of analytic over that contour.