Initial value theorem

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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). The initial value theorem then says[2]

Proof[edit]

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[edit]

Notes[edit]

  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.
  3. ^ Robert H., Jr. Cannon (4 May 2012). Dynamics of Physical Systems. Courier Dover Publications. p. 569. ISBN 978-0-486-13969-2.