In mathematics, Osgood's lemma, introduced by William Fogg Osgood (1899), is a proposition in complex analysis. It states that a continuous function of several complex variables that is holomorphic in each variable separately is holomorphic. The assumption that the function is continuous can be dropped, but that form of the lemma is much harder to prove and is known as Hartogs' theorem.
References
Osgood, William F. (1899), "Note über analytische Functionen mehrerer Veränderlichen" (PDF), Mathematische Annalen, Springer Berlin / Heidelberg, 52: 462–464, doi:10.1007/BF01476172, ISSN 0025-5831
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