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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.