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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?

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Might this be what you're looking for?

To your second question, the answer is that the radius of convergence is the distance from the center to the point nearest to the center where the function is too ill-behaved to be holomorphic.

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  • $\begingroup$ 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? $\endgroup$ – ok_ Feb 19 '15 at 1:48
  • $\begingroup$ And after some reading, this is definitely what I'm looking for! Thanks. $\endgroup$ – ok_ Feb 19 '15 at 1:55

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