Hartogs's extension theorem

Hartogs's extension theorem is a result in the theory of functions of several complex variables about the singularities of holomorphic functions. Informally it says that the set of singularities of such a function cannot be compact, so a singular set must, loosely speaking, go off to infinity in some direction. In its formal statement, if f is holomorphic on G minus K, where G is an open subset of complex n-space with n at least 2 and K is a compact subset of G whose complement in G is connected, then f extends to a unique holomorphic function on all of G. A first version was proved by Friedrich Hartogs in 1906 using Cauchy's integral formula in several variables, and the result is also called Hartogs's lemma, Hartogs's principle or the Osgood-Brown theorem. The statement fails in one variable, where 1/z is holomorphic away from 0 but cannot be continued across it, which is a basic way in which several complex variables differ from one.