In the theory of functions of several complex variables, Hartogs's 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 an isolated singularity is always a removable singularity for any analytic function of complex variables. A first version of this theorem was proved by Friedrich Hartogs,See the original paper of and its description in various historical surveys by , and . In particular, in this last reference on p. 132, the Author explicitly writes :-"As it is pointed out in the title of , and as the reader shall soon see, the key tool in the proof is the Cauchy integral formula". and as such it is known also as Hartogs's lemma and Hartogs's principle: in earlier Soviet literature,See for example , which refers the reader to the book of for a proof (however, in the former reference it is incorrectly stated that the proof is on page 324). it is also called Osgood–Brown theorem, acknowledging later work by Arthur Barton Brown and William Fogg Osgood.See and . This property of holomorphic functions of several variables is also called Hartogs's phenomenon: however, the locution "Hartogs's phenomenon" is also used to identify the property of solutions of systems of partial differential or convolution equations satisfying Hartogs type theorems.See and .
