Singularity
of an analytic function
A set of singular points (cf. Singular point) of an analytic function in the complex variables z=(z_1,\ldots,z_n), n\geq1, defined by some supplementary conditions. In particular, isolated singular points (cf. Isolated singular point) are sometimes called isolated singularities.
A set K\subset\mathbf C^n such that in a domain D adjoining K there is defined a single-valued analytic function f(z) for which the question arises of the possibility of analytic continuation of f(z) to K. For example, let D be a domain of the space \mathbf C^n, let K be a compactum contained in D, and let f(z) be holomorphic on D\setminus K. K is then a possible singularity of f(z), and the question of analytic continuation (perhaps under certain supplementary conditions) of f(z) onto the entire domain D arises; in other words, the question of "elimination" or "removal" of the singularity K.
See also Removable set.
Comments
For references see also Singular point of an analytic function and Extension theorems (in analytic geometry). See also Hartogs theorem.
Singularity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Singularity&oldid=34016