Contour Integral around a Simple Pole
The function is analytic in the entire plane, except for a simple pole at . The function is to be integrated counterclockwise over a unit circle, shown in red, which you can move in the complex plane. If the singular point falls outside the contour of integration, the function is analytic everywhere on and inside the contour and the integral equals zero by Cauchy's theorem: . When the singularity lies within the contour, the residue theorem applies and the integral equals 1. In the intermediate case, when the simple pole lies on the contour, it can be considered to be half inside, half outside. The Cauchy principal value for this segment of the integral is implied, so that the complete integral equals .
Snapshot 1: Cauchy integral formula with residue 1
Snapshot 2: Cauchy's theorem gives 0
Snapshot 3: singularity lies on contour; integral equals