Degenerating behavior of Green's function

Let the unions of real intervals $I={\cup }_{j=1}^l[a_{2j-1},a_{2j}],$ $a_1<\mathrm{.}\mathrm{.}\mathrm{.}<a_{2l},$ and $I_n={\cup }_{k=1}^m[B_{k,n},C_{k,n}]$ be such that ${\cap }_{k=1}^{\mathrm{\infty }}[B_{k,n},C_{k,n}]=\{c_k\}$ for $k=1,\mathrm{.}\mathrm{.}\mathrm{.},m$ and $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(E,I_n)\ge const>0.$ We show how to express asymptotically the Green’s function $\varphi (z,\mathrm{\infty },E\cup I_n)$ of $E\cup I_n$ at $z=\mathrm{\infty }$ in terms of the Green’s function $\varphi (z,\mathrm{\infty },E)$ and $\varphi (z,c_k,E)\mathrm{.}$ The formula yields immediately asymptotics for ${\varphi }^n(z,\mathrm{\infty },E\cup I_n)$ with respect to $n$ which are important in many problems of approximation theory. Another consequence is an asymptotic representation of $cap(E\cup I_n)$ in terms of $cap(E)$ and $\varphi (z,c_k,E)$ and of the harmonic measure $\omega (\mathrm{\infty },E_j,E\cup I_n)\mathrm{.}$