In mathematics, Hartogs's theorem is a fundamental result of Friedrich Hartogs in the theory of several complex variables. Roughly speaking, it states...

domain. That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. Holomorphic functions are...

1-\varepsilon <|z_{2}|\}\ (0<\varepsilon <1)} Hartogs's extension theorem (1906); Let f be a holomorphic function on a set G \ K, where G is a bounded (surrounded...

^{n}}|f|.} Hardy space Hardy space Hartogs 1.  Hartogs extension theorem 2.  Hartogs's theorem on separate holomorphicity harmonic A function is harmonic...

to both z (for fixed s) and s (for fixed z), and, thus, holomorphic on C × C by Hartogs' theorem. Hence, the following decomposition γ ( s , z ) = z s Γ...

but that form of the lemma is much harder to prove and is known as Hartogs' theorem. There is no analogue of this result for real variables. If it is assumed...

Hartog's theorem holds for Gateaux holomorphic functions in the following sense: If f : (U ⊂ X1) × (V ⊂ X2) → Y is a function which is separately Gateaux...