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From Wikipedia, the free encyclopedia
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]
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]
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 .
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