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$\mathbf {The \ Problem \ is}:$ Question no. 13 enter image description here

$\mathbf {My \ approach}:$ 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 .

A small hint will be helpful, thanks in advance .

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