Let $E$ be an infinite compact subset of the complex plane $\mathbb{C}$ such that $\overline{\mathbb{C}}\setminus E$ is simply connected. There exists a unique exterior conformal mapping $\Phi$ from $\overline{\mathbb{C}}\setminus E$ onto $\overline{\mathbb{C}}\setminus \{w\in \mathbb{C}: |w|\leq 1\}$ satisfying $\Phi(\infty)=\infty$ and $\Phi'(\infty)>0.$ Let $\rho_1,\rho_2\in(1,\infty)$ and $\rho_2>\rho_1.$ Set $$\Gamma_\rho=\{z\in \mathbb{C}:|\Phi(z)|=\rho\}$$
I would like to know how to prove that $$\frac{1}{(2\pi i)^2}\int_{\Gamma_{\rho_2}}\int_{\Gamma_{\rho_1}} \frac{ \Phi^{n}(t)\Phi'(z)}{(t-z)\Phi^{n+1}(z)}dz dt=1$$
Note that the function $\Phi$ is defined only outside of $E.$