Let us teach this generalization of the final-value theorem

A suggestion relevant to teaching the use of Laplace transforms in a basic course of engineering mathematics (or circuit theory, automatic control, etc) is made. The useful 'final-value' theorem for a function f(t), f (infinity) = lim sF(s), s --> 0, makes sense only if f (infinity) = lim f (t), t --> infinity, exists. A generalization of this theorem for time functions for which f (infinity) does not exist, but the time average (f) exists, states that as s --> 0, lim s F (s) = (f). This generalization includes the case of periodic or asymptotically periodic functions, and almost-periodic functions that can be given by finite sums of periodic functions. The proofs include the case of f (t) tending to f(as)(t) exponentially, which is realistic for the main physics and circuit applications. Extension of the results to discrete sequences, treatable by the z-transform, is briefly considered. The generalized form of the final-value theorem should be included in courses of engineering mathematics. The teacher can introduce interesting new problems into the lesson, and provide a better connection with the (usually later) study of the Fourier series and Fourier transform.