# Singularity

*of an analytic function*

A set of singular points (cf. Singular point) of an analytic function in the complex variables , , defined by some supplementary conditions. In particular, isolated singular points (cf. Isolated singular point) are sometimes called isolated singularities.

A set such that in a domain adjoining there is defined a single-valued analytic function for which the question arises of the possibility of analytic continuation of to . For example, let be a domain of the space , let be a compactum contained in , and let be holomorphic on . is then a possible singularity of , and the question of analytic continuation (perhaps under certain supplementary conditions) of onto the entire domain arises; in other words, the question of "elimination" or "removal" of the singularity .

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.

**How to Cite This Entry:**

Singularity.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Singularity&oldid=18225