View source for Removable singular point

{{MSC|30-XX|35-XX}}
{{TEX|done}}

The term is used often in the theory of [[Analytic function|analytic functions]] of one complex variable and sometimes in the theory of [[Partial differential equation|partial differential equations]].

Let $z_0\in \mathbb C$, $U$ be an open neighborhood of $z_0$ and $f:U\setminus \{z_0\}\to \mathbb C$ be an [[Analytic function|holomorphic function]]. Then $z_0$ is called a ''removable singular point'' 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|pole]]).

Cf. also [[Singular point|Singular point]]; [[Essential singular point|Essential singular point]]; [[Removable set|Removable set]].

More generally, in the literature of partial differential equations a [[Removable singularity|removable singularity]] of a solution $f$ of a certain partial differential equation (or a certain system of PDEs) is a point $x_0$ such that $f$ is defined in a punctured neighborhood of it and can be extended to the whole neighborhood keeping a certain degree of smoothness, in such a way that the extension is still a solution of the same partial differential equation in the larger domain. Since holomorphic functions are solutions of the [[Cauchy-Riemann equations]], this concept is indeed a generalization of the one above.

====References====
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|valign="top"|{{Ref|Al}}|| L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1966) {{MR|0188405}} {{ZBL|0154.31904}}
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|valign="top"|{{Ref|Ma}}|| A.I. Markushevich, "Theory of  functions  of a complex variable" ,  '''1–3''' , Chelsea (1977)  (Translated from  Russian) {{MR|0444912}}  {{ZBL|0357.30002}}
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|valign="top"|{{Ref|Sh}}||  B.V. Shabat, "Introduction of complex  analysis" , '''1–2''' , Moscow   (1976) (In Russian) {{MR|}}  {{ZBL|0799.32001}} {{ZBL|0732.32001}}   {{ZBL|0732.30001}}  {{ZBL|0578.32001}} {{ZBL|0574.30001}}  
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