This is a theorem in the Bak-Newman book of Complez analysis. It is stated:
Theorem 1: If $\sum_{n=0}^\infty a_nz^n$ has radius of convergence $R<\infty$ and, for every $n$, $a_n$ is real and $\ge 0$ then $f(z)=\sum_{n=0}^\infty a_nz^n$ has a singularity at $z=R$
I think I'm not understanding the statement correctly: the previous theorem is
Theorem 2: If $\sum_{n=0}^\infty a_nz^n$ has a positive radius of convergence $R$ then $f(z)=\sum_{n=0}^\infty a_nz^n$ has a singularity on the circle $|z|=R$
First of all, when talking about singularities at $z_0$, shouldn't we be working with an open (deleted) neigborhood of $z_0$? The way the theorem is stated doesn't make clear if we have a function defined on some open $D$ containing $\overline{D(0;R)}\setminus\{R\}$ and then the power series happens to have a singularity. I guess the question boils down to if a singularity (and what tyoe it is) can be determined by approaching it from only some directions.
Next, if there is a removable singularity, then the power series expansion "doesn't care" and cannot count as a singularity so the question is only about poles or essential singularities.
The power series $\sum \frac{z^n}{n^2}$ satisfies the conditions of he first theorem so there must be a singularity at $z=R=1$, where $\sum \frac{1}{n^2}<\infty$. Does it mean that the singularity is essential?
Can someone clarify all of this? Thanks!
Best Answer
The concept of singularity that the authors are using here is taken from
Definition 18.1: Suppose that $f$ is analytic in a disc $D$ and that $z_0\in\partial D$. Then $f$ is said to be regular at $z_0$ if $f$ can be continued analytically to a region $D_1$ with $z_0\in D_1$. Otherwise, $f$ is said to have a singularity at $z_0$.
So, this has nothing to do with removable or essential singularities, which are indeed concepts related to analytic functions defined on a deleted neighborhood of a point.