Hartogs' lemma

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Note: Terminology is quite confused: Hartogs' lemma sometimes refers to Hartogs' result on the Hartogs number, while Hartogs' theorem often refers to this result.

In mathematics, Hartogs' lemma is a fundamental result on several complex variables, showing that the concept of isolated singularity and removable singularity coincide for analytic functions of n > 1 complex variables. It is attributed to Friedrich Hartogs, but is also known as the Osgood-Brown theorem.

More precisely, on Cⁿ for n ≥ 2, any analytic function F defined on the complement of a compact set K extends (necessarily uniquely) to an analytic function on Cⁿ. The same is true for F defined only on the complement in an open ball or polydisc D of a compact subset. Therefore the singular set of a function of several complex variables must (loosely speaking) 'go off to infinity' in some direction.