This is related to this question. Tell me if this should be in the original question.
Is it possible for a complex function to have both conditions below?
Let $0<r<R$ and $a>0$ be a real number that satisfies $a+r<R$.
Condition 1: Power series expansion at $z=0$ has a radius of convergence $R$.
Condition 2: Power series expansion at $z=a$ has a radius of convergence $r$.
(Edit: It seems like I should swap R and r to make it consistent with the original question, but it might confuse people who have already read the question so I will leave it as it is.)
This is the case $r=1$,$a=2$ and $R=4$. So, the radius of convergence is the distance to the nearest singularity. By condition 2, this means that there is a singularity on $|z-a|=r$. However, by condition 1, the function is analytic in $|z|<R$. This means that the function is also analytic on $|z-a|=r$. This is a contradiction. So it is impossible.
My concern about this argument:
- Is the radius of convergence really the distance to the nearest singularity?
- Does the condition 1 really mean that the function is analytic in $|z|<R$?
I am not sure about those 2 question, so I am not really confident about this argument.
Is this correct? If not, is it possible for a complex function to have both conditions?