# Hartogs theorem

Hartogs' basic (principal, fundamental) theorem: If the function , defined in a domain , is holomorphic at every point with respect to each variable (for fixed , , ), then it is holomorphic in with respect to all variables. There exist many generalizations of this theorem to include cases when some of the variables are real, or when not all points of the domain are used or when some singularities of are permitted. For example: a) if a function , , , defined in the domain , is holomorphic in the domain , , and is holomorphic in the ball for any given , , then it is holomorphic in ; b) if a function defined on with values in the extended complex plane is rational with respect to each variable, then it is a rational function.

Hartogs' extension theorem: Let a domain have the form , where , , and let be bounded. Then any function that is holomorphic in a neighbourhood of the set , , can be holomorphically extended to .

Hartogs' theorem is also taken to be the theorem on the removability of compact singularities (if ); it is also known as the Osgood–Brown theorem .

The name Hartogs' theorem is also given to theorems on the continuous distribution of singular points if , on the analyticity of the set of singular points, and the theorem of uniform boundedness of a sequence of pointwise-bounded subharmonic functions.

The theorems 1), 1a), 2), and 4) were first proved by F. Hartogs.

#### References

[1] | F. Hartogs, "Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselber durch Reihen welche nach Potentzen einer Veränderlichen fortschreiten" Math. Ann. , 62 (1906) pp. 1–88 |

[2] | S. Bochner, W.T. Martin, "Several complex variables" , Princeton Univ. Press (1948) |

[3] | B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) |

#### Comments

A version of Hartogs' theorem 3) (or the Osgood–Brown theorem) is as follows (cf. [a1], Thm. 2.3.2): Let , , be an open set and let be a compact subset of such that is connected. Then every holomorphic function on can be holomorphically extended to .

The result on sequences of pointwise-bounded subharmonic functions mentioned in 4) is also called Hartogs' lemma.

#### References

[a1] | L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) pp. Chapt. 2.4 |

[a2] | S.G. Krantz, "Function theory of several complex variables" , Wiley (1982) |

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Hartogs theorem.

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