Isolated singularity

Examples

• The function has 0 as an isolated singularity.
• The cosecant function has every integer as an isolated singularity.

  Nonisolated singularities

• 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 which functions cannot be analytically continued around.

  Examples

• The function is meromorphic in, with simple poles in, for every. Since, every punctured disk centred in has an infinite number of singularities within, so no Laurent expansion is available for around, which is in fact a cluster point of its poles.
• The function has a singularity at 0 which is not isolated, since there are additional singularities at the reciprocal of every integer which are located arbitrarily close to 0.
• The function here defined as the Maclaurin series converges inside the open unit disk centred at and has the unit circle as its natural boundary.