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

Proofs

Proof using dominated convergence theorem and assuming that function is bounded

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

Proof using elementary calculus and assuming that function is bounded

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 .

Generalizing to non-bounded functions that have exponential order

The theorem assuming just that follows from the theorem for bounded :

Define . Then is bounded, so we've shown that . But and , so

since .

See also