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