# Singularity

*of an analytic function*

A set of singular points (cf. Singular point) of an analytic function $f(z)$ 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.

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Singularity.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Singularity&oldid=34016