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.[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

  1. http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node17.html
  2. Robert H. Cannon, Dynamics of Physical Systems, Courier Dover Publications, 2003, page 567.


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