Given a formal power series $\sum a_n z^n$ and a radius of convergence $R>0$, there are various ways to extend the function to the boundary such as
- Abel's theorem
- Fatou's lemma
- $H^\infty$ theorem.
What is an example of a function that does (almost) nowhere to the boundary? Which power series are proven to not possess an analytic continuation beyond the radius of convergence.
The craziest things what I can construct is a finite number of essential singularities using an entire function $f$, which is not a polynomial, and looking at something like $z \mapsto f(1/z)$.