I'm trying to follow Ahlfors's proof that any analytic function defined in an annulus $R_1 < |z-a| < R_2$ will have a Laurent representation. To do this, he defines two functions:
$$f_1(z) = \frac{1}{2\pi i} \int_{|\zeta-a|=r} \frac{f(\zeta) d\zeta}{\zeta-z} \text{ for $|z-a| < r < R_2$ } $$
$$f_2(z) = - \frac{1}{2\pi i} \int_{|\zeta - a|=r} \frac{f(\zeta)d\zeta}{\zeta-z} \text{ for $R_1 < r <|z-a|$}$$
and he says that it follows by Cauchy's integral theorem that $f(z) = f_1(z) + f_2(z)$. I was wondering if someone could explain to me why this is true.
Also, should I assume that $f_1$ is defined to be $0$ for $|z-a| \geq r$ and $f_2=0$ for $|z-a| \leq r$?