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.
[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] The original proof was given by Friedrich Hartogs in 1906, using Cauchy's integral formula for functions of several complex variables.
[1] Today, usual proofs rely on either the Bochner–Martinelli–Koppelman formula or the solution of the inhomogeneous Cauchy–Riemann equations with compact support.
Yet another very simple proof of this result was given by Gaetano Fichera in the paper (Fichera 1957), by using his solution of the Dirichlet problem for holomorphic functions of several variables and the related concept of CR-function:[5] later he extended the theorem to a certain class of partial differential operators in the paper (Fichera 1983), and his ideas were later further explored by Giuliano Bratti.
[6] Also the Japanese school of the theory of partial differential operators worked much on this topic, with notable contributions by Akira Kaneko.
Ehrenpreis' proof is based on the existence of smooth bump functions, unique continuation of holomorphic functions, and the Poincaré lemma — the last in the form that for any smooth and compactly supported differential (0,1)-form ω on Cn with ∂ω = 0, there exists a smooth and compactly supported function η on Cn with ∂η = ω.
The crucial assumption n ≥ 2 is required for the validity of this Poincaré lemma; if n = 1 then it is generally impossible for η to be compactly supported.
Furthermore, given any holomorphic function on G which is equal to f on some open set, unique continuation (based on connectedness of G \ K) shows that it is equal to f on all of G \ K. The holomorphicity of this function is identical to the condition ∂v = f ∂φ.
This defines F as a holomorphic function on G; it only remains to show (following the above comments) that it coincides with f on some open set.
On the set Cn \ L, v is holomorphic since φ is identically constant.
Since it is zero near infinity, unique continuation applies to show that it is identically zero on some open subset of G \ L.[10] Thus, on this open subset, F equals f and the existence part of Hartog's theorem is proved.