Example of continuous function that is analytic on the interior but cannot be analytically continued?

Question

I am looking for an example of a function $f$ that is 1) continuous on the closed unit disk, 2) analytic in the interior and 3) cannot be extended analytically to any larger set. A concrete example would be the best but just a proof that some exist would also be nice. (In fact I am not sure they do.)

I know of examples of analytic functions that cannot be extended from the unit disk. Take a lacuanary power series for example with radius of convergence 1. But I am not sure if any of them define a continuous function on the closed unit disk.

Answer 1

Let $f(z) = \sum z^n/n^2$, which is continuous and bounded on the closed unit disc but not analytic near $1$. Then consider

$$\sum f(z^n)/n^2.$$

This should have a singularity at every root of unity; and should be analytic in the interior because it is uniformly convergent.

Answer 2

I suggest this function: $$f(z)=\sum_{n=1}^\infty \frac{z^{n!}}{n^2}.$$ It converges uniformly on the closed unit disk, and the derivatives blow up as you approach any root of unity radially.

Answer 3

Here is a very concrete example:

$ g(z) = \sum_{n=0}^{\infty}\frac{z^{2^n + 1}}{2^n + 1}. $

The power series converges uniformly to a continuous function on the closed unit disk. Differentiating we obtain $g'(z) = f(z)$ with

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

This is the standard example of a function with a natural boundary. Clearly $f(x) \rightarrow +\infty$ as $x \rightarrow 1^{-}$ on the real axis. The functional equation

$ f(z) = z + f(z^2) $

shows that $f(x) \rightarrow + \infty$ as $x \rightarrow (-1)^{+}$ on the real axis, then $|f(z)| \rightarrow \infty$ as $z$ tends radially to ${\pm}i$, and so on, so that $|f(z)|$ tends to $\infty$ as $z$ tends radially to any root of unity of order $2^m$. Hence $f(z)$ has a dense set of singularities on the unit circle, and so does $g(z)$, thus $g(z)$ has the unit circle as natural boundary.

Answer 4

One can invoke Carathéodory's theorem.

If $U$ is a simply connected open subset of the complex plane with a Jordan curve as boundary then the Riemann map $f : U \to \mathbb D$ extends continuosly to the boundary and the extension is a homeomorphism $\partial U \to S^1$ at the boundary.

To obtain the sought function, it suffices to consider a simply connected open set $U\subset \mathbb C$ having a nowhere analytic Jordan curve as boundary and take the inverse of the Riemann map of $U$.

Answer 5

I found some neat stuff in Remmert's Classical topics in complex function theory.

Fabry's gap theorem gives a way to construct many examples including some already mentioned. Stated for the unit disk, it says:

If $m_1,m_2,\ldots$ is a sequence of positive integers such that $\displaystyle{\lim_{n\to\infty}}\frac{m_n}{n}=\infty$ and if $\displaystyle{f(z)=\sum_{n=1}^\infty a_nz^{m_n}}$ has radius of convergence 1, then the unit disk is the domain of holomorphy of $f$.

For example, if $p_n$ is the $n^{th}$ prime, then $$f(z)=\sum_{n=1}^\infty \frac{z^{p_n}}{n^2}$$ converges uniformly on the closed disk and is therefore continuous. It is not analytically extendable to any larger set because it satisfies the hypotheses of Fabry's theorem.

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An interesting result that yields many such functions in a nonconstructive way is a theorem of Fatou-Hurwitz-Pólya:

If $\displaystyle{f(z)=\sum_{n=0}^\infty a_n z^n}$ has radius of convergence 1, then the set of functions $$f_\epsilon(z)=\sum_{n=0}^\infty \epsilon_na_nz^n$$ for $\epsilon_n\in\{\pm1\}$ whose domain of holomorphy is the unit disk has cardinality $2^{\aleph_0}$.

Hausdorff showed further that if $\displaystyle{\lim_{n\to\infty} |a_n|^{1/n}}$ exists (and equals 1) then the set of such functions whose domain of holomorphy is not the unit disk is at most countable. This applies in particular to the function $\displaystyle{f(z)=\sum_{n=1}^\infty \frac{z^n}{n^2}}$, which therefore yields examples by changing the signs of the coefficients in all but countably many ways.

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One more, this time an explicit example from Remmert: The series $$f(z)=1+2z+\sum_{n=1}^\infty\frac{z^{2^n}}{2^{n^2}}$$ is one-to-one and has real derivatives of all orders on the closed disk, and has the open disk as domain of holomorphy.

Reference: Remmert's Classical topics in complex function theory, pages 252-258. (Fatou-Hurwitz-Pólya is stated on a page without preview.)