Winding number version of Cauchy integral formula

Statement

Suppose $U$ is an open subset of $C$ and $f : U \to C$ is a holomorphic function.

Suppose $c$ is a sum of oriented piecewise smooth loops $c_{1}, c_{2}, \dots, c_{r}$ , each completely inside $U$ . In other words, when we use the symbol $\oint_{c} g (z) d z$ , we actually mean $\sum_{1}^{r} \int_{c_{r}} g (z) d z$ .

Denote by $n (c; z_{0})$ the sum of the winding numbers of the $c_{k}$ about $z_{0}$ . Suppose $c$ is zero-homologous, i.e. $n (c; z) = 0$ for $z \in C ∖ U$ . Then we have, for any $z_{0} \in U$ :

$n (c; z_{0}) f (z_{0}) = \frac{1}{2 π i} \oint_{c} \frac{f (z)}{z - z_{0}} d z$

Proof

The idea here is to define analytic functions on two open subsets, show that they agree on the overlap, and hence obtain an analytic function on the whole of $C$ .

Define first:

$φ (z, w) = \frac{f (z) - f (w)}{z - w}$ if $z \neq w$ , $f^{'} (z)$ if $z = w$

Observe that $φ$ is analytic in each variable. To see this, note that for fixed $z$ , $φ$ is analytic in $w$ for $w$ away from $z$ , and the power series shows that it is analytic in a neighborhood of $z$ .

Now define an open subset:

$V : = {z \in C ∖ c ∣ n (c; z) = 0}$

Then $U \cup V = C$ , and $V$ is an open subset. Consider the function $g_{1}, g_{2} : U, V \to C$ :

$g_{1} (z) : = \oint_{c} φ (z, w) d z$

and:

$g_{2} (z) : = \oint_{c} \frac{f (w)}{w - z} d w$

To see that $g_{1}$ and $g_{2}$ agree on the intersection, observe that on the intersection $U \cap V$ , the difference is given by:

$\oint_{c} \frac{f (z)}{z - w} d z$

which is zero for points in the intersection, as they are points in $U$ about which the winding number is zero.