On the definition of (ir)regular singular points in ODE

Question

I have a simple question about the definition of singular (ir)regular point in Fuchsian theory.

For example, the Wikipedia page for Fuchsian theory consider two cases:

1. regular singular point: $f^{\prime\prime}+\dfrac{1}{z}f^\prime+\dfrac{1}{z^2}f=0$

2. irregular singular point: $f^{\prime\prime}+\dfrac{1}{z^2}f^\prime+f=0$

What about the case:

$f^{\prime\prime}+f^\prime+\dfrac{1}{z^2}f=0$

is it regular?

Answer 1

Yes, $z=0$ is a regular singular point for $f^{\prime\prime}+f^\prime+\dfrac{1}{z^2}f=0$.

I would say $z=0$ is a regular sigular point for $f^{\prime\prime}(z)+\alpha(x)f^\prime(z)+\beta(z)f(z)=0$ iff $\alpha$ has (at worst) a pole of order $1$ and $\beta$ has (at worst) a pole of order $2$ at $z=0$.