Removable singularity

A removable singularity is an isolated singularity of a holomorphic function where assigning a suitable value at the point makes the function holomorphic there.

Riemann's theorem on removable singularities: A theorem characterizing isolated singularities across which a holomorphic function extends holomorphically. It gives the precise boundedness criterion for removing the singularity.
Pole (complex analysis): An isolated singularity where a function diverges with a finite-order principal part. Unlike a removable singularity, a pole cannot be repaired by assigning a finite value.
Sinc function: The function sin z divided by z, extended to equal 1 at zero. Its apparent singularity at zero is removed by assigning the limit value.
Analytic continuation: Extension of an analytic function beyond its original domain while preserving its local analytic form. Removing a singular point is a local instance of extending an analytic function.
Laurent series: A representation of a holomorphic function on an annulus as a series with positive and negative powers. The singularity is removable exactly when all negative-power terms vanish.
Essential singularity: An isolated singularity whose Laurent expansion has infinitely many negative-power terms. It contrasts with the absent principal part at a removable singularity.
(e^z − 1)/z: A complex function with a removable singularity at zero and extension value 1. The exponential series directly exposes the finite value hidden at zero.
Identity theorem: A theorem stating that holomorphic functions agreeing on a set with an accumulation point agree throughout a connected domain. An extension across one point remains constrained by the function's values nearby.
Bounded function: A function whose absolute value stays below a fixed finite bound on its domain. Local boundedness near an isolated singularity guarantees holomorphic extension.
Isolated singularity: A point where a complex function fails to be holomorphic, with no other such points sufficiently nearby. Removable singularities are one of the three standard isolated-singularity types.

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