Winding numbers in complex analysis and Cauchy-Goursat theorem

Question

I don't understand where the cauchy goursat theorem is used and if it is at the underlined place then why is it used there because the function is not holomorphic at z0. B. r z0 represents a circle of radius r around and the delta means the boundary

Answer 1

I got the answer. The key is to take the integral of the domain with the circle around z0 removed that's why the integral is 2pi and not zero because this is a corollary of the cauchy goursat theorem.

Answer 2

Define$$\begin{array}{rccc}\eta\colon&[0,2\pi]&\longrightarrow&\mathbb C\\&t&\mapsto&z_0+re^{it}.\end{array}$$Then the paths $\gamma$ and $\eta$ are homotopic and therefore, by the Cauchy-Goursat theorem (by this I mean theorem 1.10.7 from these notes),$$\int_\gamma\frac{\mathrm dz}{z-z_0}=\int_\eta\frac{\mathrm dz}{z-z_0}.$$