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.

**References**

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

© 1996-9

1999-05-26