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

$F(s) = \int_0^\infty f(t) e^{-st}\,dt$

be the (one-sided) Laplace transform of ƒ(t). The initial value theorem then says^[2]

$\lim_{t\to 0}f(t)=\lim_{s\to\infty}{sF(s)}. \,$

Proof

Based on the definition of Laplace transform of derivative we have:

$sF(s)=f(0^-)+\int_{t=0^-}^{\infty}e^{-st}f^{'}(t)dt$

thus:

$\lim_{s \to \infty} sF(s)=\lim_{s \to \infty} [f(0^-)+\int_{t=0^-}^{\infty}e^{-st}f^{'}(t)dt]$

But $\lim_{s \to \infty}e^{-st}$ is indeterminate between t=0⁻ to t=0⁺; to avoid this, the integration can be performed in two intervals:

$\lim_{s \to \infty} [\int_{t=0^-}^{\infty}e^{-st}f^{'}(t)dt] =\lim_{s \to \infty}\{\lim_{\epsilon \to 0^+}[\int_{t=0^-}^{\epsilon}e^{-st}f^{'}(t)dt] + \lim_{\epsilon \to 0^+}[\int_{t=\epsilon}^{\infty}e^{-st}f^{'}(t)dt]\}$

In the first expression,

$e^{-st} = 1 \qquad \mathrm{when} \qquad 0^- < t < 0^+.$

In the second expression, the order of integration and limit-taking can be changed. Also

$\lim_{s \to \infty}e^{-st}(t) = 0 \qquad \mathrm{where} \qquad 0^+ < t < \infty.$

Therefore:^[3]

$\begin{align} \lim_{s \to \infty} [\int_{t=0^-}^{\infty}e^{-st}f^{'}(t)dt] &=\lim_{s \to \infty}\{\lim_{\epsilon \to 0^+}[\int_{t=0^-}^{\epsilon}f^{'}(t)dt]\} + \lim_{\epsilon \to 0^+}\{\int_{t=\epsilon}^{\infty}\lim_{s \to \infty}[e^{-st}f^{'}(t)dt]\}\\ &=f(t)|_{t=0^-}^{t=0^+} + 0\\ &= f(0^+)-f(0^-)+0\\ \end{align}$

By substitution of this result in the main equation we get:

$\lim_{s \to \infty} sF(s)=f(0^-)+f(0^+)-f(0^-)=f(0^+)$

See also

• Final value theorem

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