Let 
be a sequence of functions, each regular in a region 
, let 
for every 
and 
in 
, and let 
tend to a limit as 
at a set of points having a limit
point inside 
. Then 
tends uniformly to a limit in any region bounded by a
contour interior to 
, the limit therefore being an analytic function of 
.
Vitali's Convergence Theorem
See also
Montel's TheoremExplore with Wolfram|Alpha
References
Titchmarsh, E. C. The Theory of Functions, 2nd ed. Oxford, England: Oxford University Press, p. 168, 1960.Referenced on Wolfram|Alpha
Vitali's Convergence TheoremCite this as:
Weisstein, Eric W. "Vitali's Convergence Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VitalisConvergenceTheorem.html