High Quality Content by WIKIPEDIA articles! In mathematics, precisely in the theory of functions of several complex variables, Hartogs' extension theorem is a statement about the singularities of holomorphic functions of several variables. Informally, it states that the support of the singularities of such functions cannot be 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 the concept of isolated singularity and removable singularity coincide for analytic functions of n > 1 complex variables. A first version of this theorem was proved by Friedrich Hartogs, and as such it is known also as Hartogs' lemma and Hartogs' principle: in earlier Soviet literature it is also known as the Osgood-Brown theorem, acknowledging later work by Arthur Barton Brown and William Fogg Osgood. This property of holomorphic functions of several variables is also called Hartogs' phenomenon: however, the locution "Hartogs' phenomenon" is also used to identify the property of solutions of systems of partial differential or convolution equations satisfying Hartogs type theorems.