This question is from Ponnusamy and Silvermann, p. 369, section on harmonic functions, and I don't know how to answer it.
If $u(z)$ is harmonic for $|z|>R$ and continuous for $|z|\geq R$ , show that for $\phi =R e^{i\phi}$ , $z =r e^{i \theta}$ ($r>R$), $$\DeclareMathOperator{\Real}{\operatorname{Re}} u(z)= -\frac{1}{2\pi} \int\limits_{0}^{2\pi} \Real \left(\frac {\rho +z } { \rho -z}\right) u\big(Re^{i\theta}\big)d\phi $$ Well, result of harmonic principles were proved for disc $|z|\leq R$ and I am not able to use them as motivation in the question.
It's my humble request to you to help me in the question