# Removable singular point

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).

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.

How to Cite This Entry:
Removable singular point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Removable_singular_point&oldid=31251
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article