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 $$f(z) =\frac{1}{2\pi i}\int_{\Gamma}\frac{f(\zeta)}{z-\zeta}$$
for all $z \in\mathbb C$ with $|z| > 2.$
I tried to define a function $g(z)=f(1/z)$ and apply Cauchy Integral Formula on $g$, but could not come up with anything.