Let $\Gamma$ denote the positively oriented circle of radius $2$ with center at the origin. Let $f$ be an analytic function on $\{z\in\mathbb{C}:|z|>1\}$, and let $$\lim_{z\to\infty} f(z)=0.$$
Prove that
$$f(z)=\dfrac{1}{2\pi i}\int_{\Gamma}\dfrac{f(\zeta)}{z-\zeta}d\zeta$$,
for all $z\in\mathbb{C}$ with $|z|>2$.
I am quite bad at complex analysis and I would appreciate any hints/directions here. Also, if there are generic methods to attack such problems, please share them as well.
Thanks in advance!