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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.[1]
It is also known under the abbreviation IVT.
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[2]
Proof
Suppose first that is bounded. Say . A change of variable in the integral
shows that
- .
Since is bounded, the Dominated Convergence Theorem shows 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
See also
Notes