I was looking at the derivation for the laurent series, but didn't catch why they could use Cauchy's integral formula in their proof...
I can see that the integral of a function around the boundaries of an annulus most be zero if the function is analytic in the domain of the annulus. And so $\oint_{c_1} f(z)dz+\oint_{c_2}f(z)dz=0$, if $c_1$ is the outer boundary counter-clockwise, and $c_2$ the inner boundary clockwise. The thing I don't understand is that these can be written as:
$\oint_{c_1} f(z)dz+\oint_{c_2}f(z)dz=\frac{1}{2\pi i}\oint_{c_1}\frac{f(z)}{z-z_0}dz+\frac{1}{2\pi i}\oint_{c_2}\frac{f(z)}{z-z_0}dz$,
Using the Cauchy's integral formula, which states that that the domain must be simply connected, which an annulus isn't, if I'm not mistaken. What am I missing?