I was trying to follow a proof of Laurent's Expansion Theorem from here: Proof of Laurent series co-efficients in Complex Residue
However, I did not understand the following step:
Let () be analytic in a region bounded by two concentric circles 1 and 2 where the radius of 1 is greater than the radius of 2. By Cauchy's integral formula, we are able to show that:
$$ f(z_0) = \frac{1}{2i\pi}\oint_{C_1}\frac{f(z)}{z-z_0}dz-\frac{1}{2i\pi}\oint_{C_2}\frac{f(z)}{z-z_0}dz $$
How is this evident from Cauchy's integral theorem, I don't see it as obvious. Is it necessary to add two lines from the inter circle to the outer to "split" the annulus into different regions?