We have ,
Let $K$ be a compact subset of $G$;then there are straight line segments $\gamma_1,.....\gamma_n$ in $G-K$ such that for every function $f$ in $H(G)$(set of all holomorphic functions),
$$f(z)=\sum_{k=1}^{n}\frac{1}{2\pi i}\int_{\gamma_k}\frac{f(w)}{w-z}$$ for all $z$ in $K$.The line segments form a finite number of closed polygons.
$\textbf{Proof}$: Observe that by enlarging $K$ a little we may assume that $K=\overline{(int K)}$.Let $0<\delta<\frac{1}{2}d(K,\mathbb{C}-G)$ and place a "grid" of horizontal and vertical lines in the plane such that consecutive lines are less than a distance $\delta$ apart.Let $R_1,....,R_m$ be the resulting rectangles that intersect $K$.Also let $\partial R_j$ be the boundary of $R_j$,$1\leq j\leq m$.considered as a polygon with the counter-clockwise direction.
$\hspace{2cm}$if $z\in R_j$,$1\leq j\leq m$,then $d(z,K)<\sqrt2\delta$ so that $R_j\subset G$ by the choice of $\delta$.Also,many of the sides of the rectangle $R_1,....,R_m$ will intersect.Suppose $R_j$ and $R_i$ have a common side and let $\sigma_i$ and $\sigma_j$ be the line segments in $\partial R_i$ and $\partial R_j$ respectively,such that $R_i\cap R_j=\{\sigma_i\}=\{\sigma_j\}$.From the direction given $\partial R_i$ and $\partial R_j$,$\sigma_i$ and $\sigma_j$ are directed in the opposite sense. So if $\phi$; is any continuous function on $\{\sigma_j\}$,
$$\int_{\sigma_j}\phi+\int_{\sigma_i}\phi=0$$
let $\gamma_1,.....\gamma_n$ be those directed line segments that constitute a side of exactly one of the $R_j,1\leq j\leq m$. Thus
$$\sum_{k=1}^{n}\int_{\gamma_k}\phi=\sum_{j=1}^{m}\int_{\partial R_j}\phi......(1)$$
for every continuous function $\phi$ on $\cup_{j=1}^{m}\partial R_j.$
We claim that each $\gamma_{k}$ is in $G-K$.In fact, if one of the $\gamma_{k}$ intersects $K$, it is easy to see that there are two rectangles in the grid with $\gamma_k$ as a side and so both meet $K$. That is,$\gamma_k$ is the common side of two of the rectangles $\gamma_1,.....\gamma_n$ and this contradicts the choice of $\gamma_k$.
if $z$ belongs to $K$ and is not the boundary point of any $R_j$ then
$$\phi(w)=\frac{1}{2\pi i}(\frac{f(w)}{w-z})$$.....(2)
is continuous on $\cup_{j=1}^{m}\partial R_j$ for $f \in H(G)$.
It is difficult to visualize, and it is not complete proof, I have some questions regarding the above proof which I am finding,
(1) why we have assumed that $K=\overline{(int K)}$
(2) what is the meaning of the line let $\gamma_1,.....\gamma_n$ be those directed line segments that constitute a side of exactly one of the $R_j,1\leq j\leq m$. I am not able to visualize this line properly.
(3) How we have defined equations (1) and (2)
please if someone can clear my these doubts that will be great help for me thanks.