Residue Theorem for a self-intersecting closed curve?

Question

What does the residue theorem say about a closed curve curve as shown in this figure: figure

It seems to me that this curve self intersect at origin.

It's related to the Wick rotation and I can't close the $[-\infty, \infty]$ contour in a proper way so that the residues inside are zero, please see link for description of poles and contour.

Thanks for your answers :)

Answer 1

Split the contour into 2 contours, each a simple closed curve enclosing an arc within a quadrant of the complex plane.