-1
$\begingroup$

Let $f : D \rightarrow C$ be holomorphic on a disk $D$ and let $R(z_{j})$ be the radius of convergence of the Taylor expansion of $f(z)$ about a point $z_{j} \in D$. Show $R(z)$ is continuous on $D$.

$\endgroup$
0
$\begingroup$

Wlog assume $f$ cannot be extended holomorphically on a larger concentric disc $D_1$ - otherwise if $f$ can be extended to be entire, the radius of convergence is infinity at any point, and if not we use a maximal disc $D_1$ where $f$ can be extended, and prove the result there hence on $D$.

So $f$ has some singularities on the boundary of $D$ and let $E$ the non empty set of them. Since if $f$ can be extended holomorphically at a point on the boundary, it obviously is extended on a small arc containing that point, it follows that the complement of $E$ is open, hence $E$ is closed, hence compact. Then the required radius at a point $z$ in $D$ is obviously the distance from $z$ to $E$ (remembering that a holomorphic function on an open disc has a radius of convergence at the center of the disc at least the radius of the disc) and that is a continuos function by elementary topology.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.