nonessential singularity | regular singularity

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 either P(x) or Q(x) diverges as x->x_0 but (x - x_0) P(x) and (x - x_0)^2 Q(x) remain finite as x->x_0, then x = x_0 is called a regular singular point (or nonessential singularity).

irregular singularity | singular point