Removable singularity

Definition

Let $c$ be an isolated singularity 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 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 of $f$ at $c$ is $0$. Riemann's theorem 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.