Removable singular point
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 , 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=31251