# Analytic Extension of a function on $D^2$ with a power series representation with radius of convergence $1.$

$$\mathbf {The \ Problem \ is}:$$ Question no. 13

$$\mathbf {My \ approach}:$$ Actually, at first I assumed that each pt. in $$S^1$$ is regular point of $$f.$$

Then for each $$\alpha \in \Lambda,$$ we have $$z_{\alpha} \in S^1$$, such that there exists an analytic $$g_{\alpha}$$ on some open nbhd $$U_{\alpha}$$ (of $$z_{\alpha}$$) $$\cup D^2$$ with $$g_{\alpha}=f$$ on $$D^2 \cap U_{\alpha}.$$

Then, we have an open cover and hence a finite subcover for $$S^1.$$

But, I can't approach after this and I am unable to use the fact that $$R=\limsup |a_n|^{1/n} =1.$$

I found a proof of Hadamard's gap theorem in P.Dienes' book but this special case was not done there .