Difference between revisions of "Removable singular point"
(Importing text file) |
|||
| (One intermediate revision by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | + | {{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==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | |valign="top"|{{Ref|Al}}|| L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1966) {{MR|0188405}} {{ZBL|0154.31904}} | ||
| + | |- | ||
| + | |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}} | ||
| + | |- | ||
| + | |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}} | ||
| + | |- | ||
| + | |} | ||
Latest revision as of 08:39, 15 January 2014
2020 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 |
Removable singular point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Removable_singular_point&oldid=12605