Hartogs’ Theorem: separate analyticity implies joint
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(The present proof of this old result roughly follows the proof given in Hörmander's An Introduction to Complex Analysis in Several Variables, which I believe roughly follows Hartogs' original argument.) Theorem: Let f be a C-valued function defined in an open set U ⊂ C n. Suppose that f is analytic in each variable z j when the other coordinates z k for k = j are fixed. Then f is analytic as a function of all n coordinates. Remark: Absolutely no additional hypothesis on f is used beyond its separate analyticity. Specifically, there is no assumption of continuity, nor even of measurability. Indeed, the beginning of the proof illustrates the fact that an assumption of continuity trivializes things. The strength of the theorem is that no hypothesis whatsoever is necessary. Proof: The assertion is local, so it suffices to prove it when the open set U is a polydisk. The argument approaches the full assertion in stages. First, suppose that f is continuous on the closure ¯ U of a polydisk U , and separately analytic. Even without continuity, simply by separate analyticity, an n-fold iterated version of Cauchy's one-variable integral formula is valid, namely f (z) = 1 (2πi) n C1. .. where C j is the circle bounding the disk in which z j lies, traversed in the positive direction. The integral is a compactly supported integral of the function For |z j | < |ζ j |, the geometric series expansion 1 ζ j − z j = n≥0 z n j ζ n+1 j can be substituted into the latter integral. Fubini's theorem justifies interchange of summation and integration, yielding a (convergent) power series for f (z). Thus, continuity of f (z) (with separate analyticity) implies joint continuity. Note that if we could be sure that every conceivable integral of analytic functions were analytic, then this iterated one-variable Cauchy formula would prove (joint) analyticity immediately. However, it is not obvious that separate analyticity implies continuity, for example. Next we see that boundedness of a separately analytic function on a closed polydisk implies continuity, using Schwarz' lemma and its usual corollary: Lemma: (Schwarz) Let g(z) be a holomorphic function on {z ∈ C : |z| < 1}, with g(0) = 0 and |g(z)| ≤ 1.
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تاریخ انتشار 2005