[Math] Power series without analytic continuation

complex-analysispower series

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 $z \mapsto f(1/z)$.

Best Answer

This is problem 2 in Chapter 2 of Stein & Shakarchi's Complex Analysis. They give two examples:

$f(z) = \sum_{n=0}^\infty z^{2^n},$

and, for $0 < \alpha < \infty,$

$f(z) = \sum_{n = 0}^\infty 2^{-n \alpha} z^{2^n}.$

The latter can in fact be extended continuously but not analytically to the boundary circle.

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