Natural boundary

Remarks

The most common case is a power series with radius of convergence $R$, with $U$ its open disc of convergence. On the circle of convergence there is always at least one point that is not regular. Example: $f(z) = \sum_{n \ge 0} z^{2^n}$ on the unit disc has the unit circle as natural boundary. If $\zeta^{2^k} = 1$, then $f(r\zeta) = \sum_{n < k} (r\zeta)^{2^n} + \sum_{n \ge k} r^{2^n} \to \infty$ as $r \to 1^-$. Such $\zeta$ are dense in the circle, while $f$ would be bounded near a regular point. Non-example: $\frac{1}{1 - z} = \sum_{n \ge 0} z^n$ does not have the unit circle as natural boundary. Each point $\zeta \neq 1$ with $\abs{\zeta} = 1$ is regular, since $\frac{1}{1 - z}$ is holomorphic on a disc around $\zeta$.