Isolated singularity

Nonisolated singularities

Other than isolated singularities, complex functions of one variable may exhibit other singular behavior. Namely, two kinds of nonisolated singularities exist:

• Cluster points, i.e. limit points of isolated singularities: if they are all poles, despite admitting Laurent series expansions on each of them, no such expansion is possible at its limit.
• Natural boundaries, i.e. any non-isolated set (e.g. a curve) around which functions cannot be analytically continued (or outside them if they are closed curves in the Riemann sphere).

Examples

• The function $\tan \left(\frac{1}{z}\right)$ is meromorphic on ⁠${\displaystyle \mathbb{C}\setminus \{0\}}$⁠, with simple poles at ⁠${\displaystyle z_n={\left(\frac{\pi }{2}+n\pi \right)}^{-1}}$⁠, for every ⁠${\displaystyle n\in {\mathbb{N}}_0}$⁠. Since ⁠${\displaystyle z_n\to 0}$⁠, every punctured disk centered at ${\displaystyle 0}$ has an infinite number of singularities within it, so no Laurent expansion is available for ⁠${\displaystyle \tan \left(\frac{1}{z}\right)}$⁠ around ⁠${\displaystyle 0}$⁠, which is in fact a cluster point of its poles.
• The function ⁠${\displaystyle \csc \left(\frac{\pi }{z}\right)}$⁠ has a singularity at ⁠${\displaystyle 0}$⁠ that is not isolated, since there are additional singularities at the reciprocal of every integer, which are located arbitrarily close to ⁠${\displaystyle 0}$⁠ (though the singularities at these reciprocals are themselves isolated).
• The function defined via the Maclaurin series ⁠${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{z}^{2^n}}$⁠ converges inside the open unit disk centred at ⁠${\displaystyle 0}$⁠ and has the unit circle as its natural boundary.