Singular Point (Differential Equation)

Consider a second-order Ordinary Differential Equation

$$y''+P(x)y'+Q(x)y=0.$$

If $P(x)$ and $Q(x)$ remain Finite at $x=x_0$, then $x_0$ is called an Ordinary Point. If either $P(x)$ or $Q(x)$ diverges as $x\to x_0$, then $x_0$ is called a singular point. Singular points are further classified as follows:

1. If either $P(x)$ or $Q(x)$ diverges as $x\to x_0$ but $(x-x_0)P(x)$ and $(x-x_0)^2Q(x)$ remain Finite as $x\to x_0$, then $x=x_0$ is called a Regular Singular Point (or Nonessential Singularity).

2. If $P(x)$ diverges more quickly than $1/(x-x_0)$, so $(x-x_0)P(x)$ approaches Infinity as $x\to x_0$, or $Q(x)$ diverges more quickly than $1/(x-x_0)^2Q$ so that $(x-x_0)^2Q(x)$ goes to Infinity as $x\to x_0$, then $x_0$ is called an Irregular Singularity (or Essential Singularity).

See also Irregular Singularity, Regular Singular Point, Singularity

References

Arfken, G. ``Singular Points.'' §8.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 451-454, 1985.