In mathematics, precisely in the theory of functions of [[several complex variables]], '''Hartogs' extension theorem''' is a statement about the [[Singularity (mathematics)|singularities]] of [[holomorphic function]]s of several variables. Informally, it states that the [[Support (mathematics)|support]] of the singularities of such functions cannot be [[compact space|compact]], therefore the singular set of a function of several complex variables must (loosely speaking) 'go off to infinity' in some direction. More precisely, it shows that an [[isolated singularity]] is always a [[removable singularity]] for any [[analytic function]]s of {{math|''n'' > 1}} complex variables. A first version of this theorem was proved by [[Friedrich Hartogs]],<ref name="hartogs">See the original paper of {{Harvtxt|Hartogs|1906}} and its description in various historical surveys by {{harvtxt|Osgood|1963|pp=56–59}}, {{harvtxt|Severi|1958|pp=111–115}} and {{harvtxt|Struppa|1988|pp=132–134}}. In particular, in this last reference on p. 132, the Author explicitly writes :-"''As it is pointed out in the title of {{harv|Hartogs|1906}}, and as the reader shall soon see, the key tool in the proof is the [[Cauchy integral formula]]''".</ref> and as such it is known also as '''Hartogs' lemma''' and '''Hartogs' principle''': in earlier [[Soviet Union|Soviet]] literature,<ref>See for example {{harvtxt|Vladimirov|1966|p=153}}, which refers the reader to the book of {{harvtxt|Fuks|1963|p=284}} for a proof (however, in the former reference it is incorrectly stated that the proof is on page 324).</ref> it is also called '''Osgood-Brown theorem''', acknowledging later work by [[Arthur Barton Brown]] and [[William Fogg Osgood]].<ref>See {{harvtxt|Brown|1936}} and {{harvtxt|Osgood|1929}}.</ref> This property of holomorphic functions of several variables is also called '''[[#Hartogs' phenomenon|Hartogs' phenomenon]]''': however, the locution "Hartogs' phenomenon" is also used to identify the property of solutions of [[System of equations|systems]] of [[partial differential equation|partial differential]] or [[convolution operator|convolution equation]]s satisfying Hartogs type theorems.<ref>See {{harvtxt|Fichera|1983}} and {{harvtxt|Bratti|1986a}} {{harv|Bratti|1986b}}.</ref>
