Definition:Removable Singularity/Complex Function

Definition

Let $U \subseteq \C$ be an open set.

Let $f : U \to \C$ be a complex function.

An isolated singularity of $f$ is a removable singularity if and only if $f$ can be extended to a holomorphic function $f: U \to \C$.

Examples

Example: $\dfrac {\sin z} z$

Let $f: \C \setminus \set 0 \to \C$ be the complex function defined as:

$\map f z = \dfrac {\sin z} z$

Then $f$ has a removable singularity at the point $z = 0$.

Also see

• Results about removable singularities can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): removable singularity
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): singular point (singularity): 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): removable singularity
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): singular point (singularity): 1.