View source for Singularity

''of an analytic function''

A set of singular points (cf. [[Singular point|Singular point]]) of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085630/s0856301.png" /> in the complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085630/s0856302.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085630/s0856303.png" />, defined by some supplementary conditions. In particular, isolated singular points (cf. [[Isolated singular point|Isolated singular point]]) are sometimes called isolated singularities.

A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085630/s0856304.png" /> such that in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085630/s0856305.png" /> adjoining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085630/s0856306.png" /> there is defined a single-valued analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085630/s0856307.png" /> for which the question arises of the possibility of [[Analytic continuation|analytic continuation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085630/s0856308.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085630/s0856309.png" />. For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085630/s08563010.png" /> be a domain of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085630/s08563011.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085630/s08563012.png" /> be a compactum contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085630/s08563013.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085630/s08563014.png" /> be holomorphic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085630/s08563015.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085630/s08563016.png" /> is then a possible singularity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085630/s08563017.png" />, and the question of analytic continuation (perhaps under certain supplementary conditions) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085630/s08563018.png" /> onto the entire domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085630/s08563019.png" /> arises; in other words, the question of  "elimination"  or  "removal"  of the singularity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085630/s08563020.png" />.

See also [[Removable set|Removable set]].



====Comments====
For references see also [[Singular point|Singular point]] of an analytic function and [[Extension theorems (in analytic geometry)|Extension theorems (in analytic geometry)]]. See also [[Hartogs theorem|Hartogs theorem]].