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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.


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:
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article