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Irregular 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 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).

See also

Ordinary Point, Regular Singular Point, Singular Point

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References

Arfken, G. "Singular Points." §8.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 451-453 and 461-463, 1985.

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Irregular Singularity

Cite this as:

Weisstein, Eric W. "Irregular Singularity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IrregularSingularity.html

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