Let 
be an analytic function of 
, regular in the half-strip 
defined by 
and 
. If 
is bounded in 
and tends to a limit 
as 
for a certain fixed value 
of 
between 
and 
, then 
tends to this limit 
on every line 
in 
, and 
uniformly for 
.
Montel's Theorem
See also
Vitali's Convergence TheoremExplore with Wolfram|Alpha
References
Krantz, S. G. "Montel's Theorem, First Version and Montel's Theorem, Second Version." §8.4.3 and 8.4.4 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 114, 1999.Titchmarsh, E. C. The Theory of Functions, 2nd ed. Oxford, England: Oxford University Press, p. 170, 1960.Referenced on Wolfram|Alpha
Montel's TheoremCite this as:
Weisstein, Eric W. "Montel's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MontelsTheorem.html