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 Laplace transform of ƒ. If is bounded on 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.