Reference:Function of one comlex variable by J.B Conway.
Let $K$ be a compact subset of the region $G$,then there are straight lines $\gamma_1,\gamma_2,\ldots,\gamma_n$ in $G-K $ such that for every function $f$ in $H(G)$,$$\sum _{k=1}^n \int_{\gamma_k}\frac{f(w)}{w-z}\,dw$$
$\forall z\in K$.The line segments form a finite number of closed polygons.
I've two minor queries-
$1.$How $\phi (w)=\frac{1}{2\pi i}\frac{f(w)}{w-z}$ happened before equation(1.3)?
$2.$How,If $z \notin R_j$,then $\frac{1}{2\pi i}\int_{\partial R_j}\frac{f(w)}{w-z}dw=0$ after equation (1.3)?