Singular Point (Function)

Singular points (also simply called ``singularities'') are points $z_0$ in the Domain of a Function $f$ where $f$ 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 $n$ is a singularity $z_0$ of $f(z)$ for which the function $(z-z_0)^nf(z)$ is nonsingular and for which $(z-z_0)^k f(z)$ is singular for $k=0$, 1, ..., $n-1$.

Removable Singularities are singularities for which it is possible to assign a Complex Number in such a way that $f(z)$ becomes Analytic. For example, the function $f(z) = z^2/z$ has a Removable Singularity at 0, since $f(z) = z$ everywhere but 0, and $f(z)$ can be set equal to 0 at $z=0$. Removable Singularities are not Poles.

The function $f(z) = \csc(1/z)$ has Poles at $z=1/(2\pi n)$, 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.