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 .
See also Vitali's Convergence Theorem
Titchmarsh, E. C. The Theory of Functions, 2nd ed. Oxford, England: Oxford University Press, p. 170, 1960.