Removable singular point

2020 Mathematics Subject Classification: Primary: 30-XX [MSN][ZBL]

of a single-valued holomorphic function $f$ of one complex variable

Let $z_0\in \mathbb C$, $U$ an open neighborhood of $z_0$ and $f:U\setminus \{z_0\}\to \mathbb C$ an holomorphic function. Then $z_0$ is called a removable singularity of the function $f$ if there is a positive radius $r_0$ such that $f$ is bounded on the punctured disk $\{z: 0<|z-z_0|<r_0\}$.

Indeed, under this condition the limit \[ w_0 := \lim_{z\to z_0} f(z) \] exists and if we extend $f$ to $U$ by setting $f(z_0) = w_0$, then the resulting extension is holomorphic on the whole open set $U$.

The condition is obviously sharp, since the map $z\mapsto z^{-1}$ is holomorphic on $\mathbb C\setminus \{0\}$ but cannot be extended to a continuous function on the whole complex plane (indeed $0$ is, in this case, a pole).

Cf. also Singular point; Essential singular point; Removable set.

References

[Al]	L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1966) MR0188405 Zbl 0154.31904
[Ma]	A.I. Markushevich, "Theory of functions of a complex variable" , 1–3 , Chelsea (1977) (Translated from Russian) MR0444912 Zbl 0357.30002
[Sh]	B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) Zbl 0799.32001 Zbl 0732.32001 Zbl 0732.30001 Zbl 0578.32001 Zbl 0574.30001