Initial value theorem

In mathematical analysis, the initial value theorem is a theorem used to relate frequency domain expressions to the time domain behavior as time approaches zero. Let be the (one-sided) Laplace transform of ƒ(t). If is bounded on (or if just ) and exists then the initial value theorem says Suppose first that is bounded, i.e. . A change of variable in the integral shows that Since is bounded, the Dominated Convergence Theorem implies that Of course we don't really need DCT here, one can give a very simple proof using only elementary calculus: Start by choosing so that , and then note that uniformly for . The theorem assuming just that follows from the theorem for bounded : Define . Then is bounded, so we've shown that . But and , so since .