7. Theorem 2.9.2 - Global Cauchy Formula

Let $G \subseteq C$ be a domain, and let $γ : [t_{0}, t_{1}] \to G$ be a closed curve in $G$ which is homologous to zero in $G$ .

Let $f : G \to C$ be holomorphic. Then for any point $z \in G - i m (γ)$ , we have:
:?:

\frac{1}{2 π i} \oint_{γ} \frac{f (ζ)}{ζ - z} d ζ = n_{γ} (z) f (z)

Remark

Recall that if we "divide out" the $f (z)$ term, we have the original analytic formula for the winding number.

Note

To prove this statement, move everything to the left-hand side and try to show that the integral is zero.