Singular Point (Function)

Singular points (also simply called singularities'') are points in the Domain of a Function where fails to be Analytic. Isolated Singularities may be classified as Essential Singularities, Poles, or Removable Singularities.

Essential Singularities are Poles of Infinite order.

A Pole of order is a singularity of for which the function is nonsingular and for which is singular for , 1, ..., .

Removable Singularities are singularities for which it is possible to assign a Complex Number in such a way that becomes Analytic. For example, the function has a Removable Singularity at 0, since everywhere but 0, and can be set equal to 0 at . Removable Singularities are not Poles.

The function has Poles at , and a nonisolated singularity at 0.

See also Essential Singularity, Irregular Singularity, Ordinary Point, Pole, Regular Singular Point, Removable Singularity, Singular Point (Differential Equation)

References

Arfken, G. Singularities.'' §7.1 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 396-400, 1985.