Montel's Theorem

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 .

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.