Removable singularity

In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourhood of that point.

For instance, the (unnormalized) sinc function, as defined by

${\displaystyle {\text{sinc}}(z)={\frac {\sin z}{z}}}$

has a singularity at ⁠${\displaystyle z=0}$⁠. This singularity can be removed by defining ⁠${\displaystyle {\text{sinc}}(0):=1}$⁠, which is the limit of sinc as ⁠${\displaystyle z}$⁠ tends to ⁠${\displaystyle 0}$⁠. The resulting function is holomorphic. In this case the problem was caused by sinc being given an indeterminate form. Taking a power series expansion for ⁠${\displaystyle \textstyle {\frac {\sin(z)}{z}}}$⁠ around the singular point shows that

${\displaystyle {\text{sinc}}(z)={\frac {1}{z}}\left(\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{(2k+1)!}} ight)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{(2k+1)!}}=1-{\frac {z^{2}}{3!}}+{\frac {z^{4}}{5!}}-{\frac {z^{6}}{7!}}+\cdots .}$

Formally, if ${\displaystyle U\subset \mathbb {C} }$ is an open subset of the complex plane⁠${\displaystyle \mathbb {C} }$⁠, ${\displaystyle a\in U}$ a point of ⁠${\displaystyle U}$⁠, and ${\displaystyle f:U\smallsetminus \{a\} ightarrow \mathbb {C} }$ is a holomorphic function, then ${\displaystyle a}$ is called a removable singularity for ${\displaystyle f}$ if there exists a holomorphic function ${\displaystyle g:U ightarrow \mathbb {C} }$ which coincides with ${\displaystyle f}$ on ⁠${\displaystyle U\smallsetminus \{a\}}$⁠. We say ${\displaystyle f}$ is holomorphically extendable over ${\displaystyle U}$ if such a ${\displaystyle g}$ exists.