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''of a single-valued [[Analytic function|analytic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081240/r0812401.png" /> of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081240/r0812402.png" />''
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{{MSC|30-XX|35-XX}}
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{{TEX|done}}
  
A term to denote a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081240/r0812403.png" /> which has a punctured neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081240/r0812404.png" /> in which the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081240/r0812405.png" /> is analytic and bounded.
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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]].
  
Under these conditions the finite limit
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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 $\{z: 0<|z-z_0|<r_0\}$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081240/r0812406.png" /></td> </tr></table>
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Indeed, under this condition the limit
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\[
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w_0 := \lim_{z\to z_0} f(z)
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\]
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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 $U$.
  
exists. This assigns a value to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081240/r0812407.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081240/r0812408.png" />, resulting in an analytic function in a full neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081240/r0812409.png" />.
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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]]).
  
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Cf. also [[Singular point|Singular point]]; [[Essential singular point|Essential singular point]]; [[Removable set|Removable set]].
  
 
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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.
====Comments====
 
Cf. also, e.g., [[Singular point|Singular point]]; [[Essential singular point|Essential singular point]]; [[Removable set|Removable set]].
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"A.I. Markushevich,   "Theory of functions of a complex variable" , '''1''' , Chelsea (1977)  (Translated from Russian)</TD></TR></table>
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{|
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|-
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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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|-
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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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|-
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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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|-
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|}

Latest revision as of 08:39, 15 January 2014

2010 Mathematics Subject Classification: Primary: 30-XX Secondary: 35-XX [MSN][ZBL]

The term is used often in the theory of analytic functions of one complex variable and sometimes in the theory of 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 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).

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

More generally, in the literature of partial differential equations a 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

[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
How to Cite This Entry:
Removable singular point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Removable_singular_point&oldid=12605
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article