Vitali's Convergence Theorem

Let f_n(z) be a sequence of functions, each regular in a region D, let |f_n(z)|<=M for every n and z in D, and let f_n(z) tend to a limit as n->infty at a set of points having a limit point inside D. Then f_n(z) tends uniformly to a limit in any region bounded by a contour interior to D, the limit therefore being an analytic function of z.

See also

Montel's Theorem

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Titchmarsh, E. C. The Theory of Functions, 2nd ed. Oxford, England: Oxford University Press, p. 168, 1960.

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Vitali's Convergence Theorem

Cite this as:

Weisstein, Eric W. "Vitali's Convergence Theorem." From MathWorld--A Wolfram Web Resource.

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