# Determining radius of convergence from singular points

Say I have a differential equation

$$P(x)y^{\prime\prime}+Q(x)y^\prime+R(x)y=0$$

and I know $P(x)=0$ at $x=\pm i$. Then there is a theorem which helps us conclude that the series solution centered at $x=0$ for $y$ converges for $|x|<|\pm i-0|=1$.

I have two questions:

1. What is the name of this theorem? I have been searching my textbook and Googling with no success.

2. Suppose $P(x)=0$ at $x=1,2$. I want my series to center at $x=0$. Then is my radius of convergence $|x|<1$ or $|x|<2$ or is there some other mess going on?

• Though $1$ and $2$ are both singular points, my radius of convergence only necessarily converges for $|x|<1$? Just to be clear I am understanding correctly. And it is still possible that I could converge for $|x|\geq1$, it's just not necessarily true? – ok_ Feb 19 '15 at 1:48