---
tag: 0P18JC72RE
kind: definition
name: Removable singularity
msc: [30D30]
defines: 09R8GC85RB
uses: [0TYB3HM9KE, 0DG0Z72CQD]
sources:
  - url: https://en.wikipedia.org/wiki/Removable_singularity
    locator: lead section
---

## Definition

Let $c$ be an [[isolated singularity|0TYB3HM9KE]] of $f$, so that $f$ is holomorphic on $0 < |z - c| < r$ for some $r > 0$.
The singularity $c$ is removable if there is a function $g$ that is [[holomorphic|0DG0Z72CQD]] on the disc $|z - c| < r$ with $g(z) = f(z)$ for each $z$ with $0 < |z - c| < r$.

## Remarks

The standard example is $f(z) = \frac{\sin z}{z}$ at $0$. With $g(0) = 1$, the function $g$ is the sum of $1 - \frac{z^2}{3!} + \frac{z^4}{5!} - \cdots$, which is holomorphic.
The singularity is removable if and only if the [[principal part|0NDV5R6Z6X]] of $f$ at $c$ is $0$.
[[Riemann's theorem|0BSQDZWF34]] says that the singularity is removable if $f$ is bounded near $c$.

## Explanations

### L1 human

A removable singularity is a hole in the domain of the function that you can fill with one value.
After you fill it, the function is holomorphic at the point too.