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Irregular Singular Point
Definition: 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)^2Q(x) goes to infinity as x->x_0, then x_0 is called an irregular singularity (or essential singularity).