Cauchy's integral formula for an exterior region

(Cauchy’s formula for “exterior” region) Let \(\gamma\) be a piecewise smooth simple closed curve with interior \(\Omega_1\) and exterior \(\Omega_2\). Assume \(f'(z)\) exists in an open set containing \(\gamma\) and \(\Omega_2\), and \[ \lim_{z\to\infty}f(z)=A. \] Show that \[ \frac{1}{2\pi i}\int_\gamma \frac{f(\xi)}{\xi-z}\,d\xi = \begin{cases} A, & z\in\Omega_1,\\ -f(z)+A, & z\in\Omega_2. \end{cases} \]