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Initial value theorem

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]

Proofs

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Proof using dominated convergence theorem and assuming that function is bounded

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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

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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

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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

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Notes

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  1. ^ Fourier and Laplace transforms. R. J. Beerends. Cambridge: Cambridge University Press. 2003. ISBN 978-0-511-67510-2. OCLC 593333940.{{cite book}}: CS1 maint: others (link)
  2. ^ Robert H. Cannon, Dynamics of Physical Systems, Courier Dover Publications, 2003, page 567.