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(∞) = lim sF(s), s → 0, makes sense only if f(∞) = lim f(t), t → ∞, exists. A generalization of this theorem for time functions for which f(∞) does not exist, but the time average exists, states that as s → 0, lim sF(s) = . 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.
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