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 (t exists then the initial value theorem says

Proofs

Proof using dominated convergence theorem and assuming that function is bounded

shows that

1. .

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:

uniformly.

Generalizing to non-bounded functions that have exponential order

:

, so

.

See also

• Final value theorem

Notes