##  [Removable Singularity](/removable-singularity-1) 

 Definition

An isolated singular point a of a complex function f where f can be redefined or extended at a so that the extended function is analytic (holomorphic) on a neighborhood of a.

 

 

 

 

 

 





## Principle

Principle

A singularity is removable precisely when the principal part of the Laurent expansion about that point vanishes (equivalently f is bounded near the point or the limit of f(z) as z→a exists); the extension, when it exists, is unique.

 

 

 

 

 





## Demonstration

Demonstration

Example: f(z)=sin(z)/z has a singularity at z=0 but its Laurent series has no negative terms beyond the (z^{-1}) term cancelation, so defining f(0)=1 yields an analytic extension at 0.

 

 

 

 

## Misapplication

Misapplication

Treating any isolated point where f is bounded along some sequences as removable; boundedness along sequences is insufficient — boundedness in a punctured neighborhood (or vanishing principal part) is required.

 

 

 

 

 





## Consequence

Consequence

When correctly identified, a removable singularity can be eliminated to produce a holomorphic function on a larger domain; this preserves analytic continuation and does not change residues because the residue is zero.

 

 

 

 

## Reversal

Reversal

Inversion produces poles or essential singularities: a pole corresponds to a genuinely infinite principal part and an essential singularity to an infinite negative tail in the Laurent series.

 

 

 

 

 





## Boundary

Boundary

Applies only to isolated singularities of single-valued analytic functions; does not apply to branch points, accumulation points of singularities, or multi-valued branch-type behaviors.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension arises with 'removable' versus 'apparently bounded but nonextendable' behaviors and with poles when limits are finite versus infinite; distinguishing boundedness in punctured neighborhoods from pointwise bounded sequences is essential.

 

 

 

 

 





## Synthesis

Synthesis

A removable singularity is an isolated point where the analytic obstruction is only a definitional hole: the Laurent principal part is absent, allowing a unique holomorphic extension that restores regularity locally.