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Singular Point (Differential Equation)

Consider a second-order Ordinary Differential Equation

\begin{displaymath}
y''+P(x)y'+Q(x)y=0.
\end{displaymath}

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.




© 1996-9 Eric W. Weisstein
1999-05-26