---
tag: T01779NV3Z
kind: definition
name: Ordinary point and singular point
msc: [34M03, 34A25]
defines: QK2J038VB4
uses: [0QQRP17Z7S, 081WKEDK6C, 0ZZ7ZYDR8W, 0J78KBPFZB]
sources:
  - url: https://en.wikipedia.org/wiki/Frobenius_method
---

## Definition

Let $U \subseteq \C$ be open, $z_0 \in U$, and let $p_1, \ldots, p_n$ be [[meromorphic functions|0ZZ7ZYDR8W]] on $U$.
Consider the homogeneous [[linear equation|0QQRP17Z7S]]
$$y^{(n)} + p_1(z)\, y^{(n-1)} + \cdots + p_n(z)\, y = 0.$$
The point $z_0$ is an *ordinary point* of the equation if every $p_k$ is [[analytic|081WKEDK6C]] at $z_0$ (after removing removable singularities).
Otherwise $z_0$ is a *singular point*: some $p_k$ has a pole at $z_0$.

## Remarks

The point $\infty$ of the [[Riemann sphere|0J78KBPFZB]] 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.