I refer to Cauchy's theorem for a disk with singularities (Theorem 5), in the book Complex Analysis by Lars:
The proof given is as follows (page 113 - 114):
I have a doubt about the proof. What happens if there is a singularity at the center of the disk $(x_0,y_0)$? How would we define the integral $F(z)$ then? Supposing that we pick some other point as the "base point", how do we ensure that we always have a well-defined rectilinear path contained within the disk? How do we prove that any two rectilinear paths from the base point to the same point have the same integral?