Initial value theorem

Last updated October 25, 2021

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]

Proof

Suppose first that $f$ is bounded. Say $\lim _{t\to 0^{+}}f(t)=\alpha$ . A change of variable in the integral $\int _{0}^{\infty }f(t)e^{-st}\,dt$ shows that

sF(s)=\int _{0}^{\infty }f\left({\frac {t}{s}}\right)e^{-t}\,dt

.

Since $f$ is bounded, the Dominated Convergence Theorem shows that

\lim _{s\to \infty }sF(s)=\int _{0}^{\infty }\alpha e^{-t}\,dt=\alpha .

Of course we don't really need DCT here, one can give a very simple proof using only elementary calculus:

Start by choosing $A$ so that $\int _{A}^{\infty }e^{-t}\,dt<\epsilon$ , and then note that $\lim _{s\to \infty }f\left({\frac {t}{s}}\right)=\alpha$ uniformly for $t\in (0,A]$ .

The theorem assuming just that $f(t)=O(e^{ct})$ follows from the theorem for bounded $f$ : Define $g(t)=e^{-ct}f(t)$ . Then $g$ is bounded, so we've shown that $g(0^{+})=\lim _{s\to \infty }sG(s)$ . But $f(0^{+})=g(0^{+})$ and $G(s)=F(s+c)$ , so

\lim _{s\to \infty }sF(s)=\lim _{s\to \infty }(s-c)F(s)=\lim _{s\to \infty }sF(s+c)=\lim _{s\to \infty }sG(s),

since $\lim _{s\to \infty }F(s)=0$

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.

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[1] ttp://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node17.html

[2] Robert H. Cannon, Dynamics of Physical Systems, Courier Dover Publications, 2003, page 567.

[1]

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Initial value theorem

Contents

Proof

See also

Notes

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