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^+)

↑ 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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