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->x_0, then x_0 is called a singular point. If P(x) diverges more quickly than 1/(x - x_0), so (x - x_0) P(x) approaches infinity as x->x_0, or Q(x) diverges more quickly than 1/(x - x_0)^2 so that (x - x_0)^2 Q(x) goes to infinity as x->x_0, then x_0 is called an irregular singularity (or essential singularity).

ordinary point | regular singular point | singular point