Hartogs's extension theorem

"Hartogs' lemma" redirects here. For the lemma on infinite ordinals, see Hartogs number.

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 n > 1 complex variables. A first version of this theorem was proved by Friedrich Hartogs,[1] and as such it is known also as Hartogs's lemma and Hartogs's principle: in earlier Soviet literature,[2] it is also called the Osgood–Brown theorem, acknowledging later work by Arthur Barton Brown and William Fogg Osgood.[3] 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.[4]

  1. ^ See the original paper of Hartogs (1906) and its description in various historical surveys by Osgood (1966, pp. 56–59), Severi (1958, pp. 111–115) and 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 (Hartogs 1906), and as the reader shall soon see, the key tool in the proof is the Cauchy integral formula".
  2. ^ See for example Vladimirov (1966, p. 153), which refers the reader to the book of Fuks (1963, p. 284) for a proof (however, in the former reference it is incorrectly stated that the proof is on page 324).
  3. ^ See Brown (1936) and Osgood (1929).
  4. ^ See Fichera (1983) and Bratti (1986a) (Bratti 1986b).