Ordinary point and singular point

Remarks

The point $\infty$ of the Riemann sphere is ordinary or singular according as $w = 0$ is ordinary or singular for the equation obtained by the substitution $z = 1/w$. For real equations with real-analytic coefficients the same words are used, with $z$ replaced by a real variable $x$. The definition applies to the normalised equation; for $a_0(z) y'' + a_1(z) y' + a_2(z) y = 0$ one first divides by $a_0$.

Example: for $y'' - 2 z y' + 4 y = 0$, the coefficients $-2z$ and $4$ are polynomials, so every $z_0 \in \C$ is an ordinary point.

Non-example: for $z^2 y'' + z y' + (z^2 - 1) y = 0$, the normalised coefficient $p_1(z) = \frac{1}{z}$ has a pole at $0$, so $0$ is not an ordinary point but a singular point.

At an ordinary point $z_0$, every solution is analytic, and its Taylor series at $z_0$ converges at least on the largest disc around $z_0$ on which all $p_k$ are analytic.