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). The initial value theorem then says[2]
Proof
Based on the definition of Laplace transform of derivative we have:
thus:
But is indeterminate between t=0− to t=0+; to avoid this, the integration can be performed in two intervals:
In the first expression,
In the second expression, the order of integration and limit-taking can be changed. Also
Therefore:[3]
By substitution of this result in the main equation we get:
See also
Notes
- ↑ http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node17.html
- ↑ Robert H. Cannon, Dynamics of Physical Systems, Courier Dover Publications, 2003, page 567.
- ↑ Robert H., Jr. Cannon (4 May 2012). Dynamics of Physical Systems. Courier Dover Publications. p. 569. ISBN 978-0-486-13969-2.
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