Initial value theorem explained

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]

Let

F(s) = \int ^{inf ty} _{0} f(t) e ^{-st} dt

be the (one-sided) Laplace transform of ƒ(t). If

f is bounded on ( 0 , inf ty) (or if just f(t) = O(e ^{ct} ) ) and \lim _{t\to 0 ^{+}} f(t) exists then the initial value theorem says^[2] \lim _{t\to} f(t) = \lim _{s\toinfty} {sF(s)}.

Proofs

Proof using dominated convergence theorem and assuming that function is bounded

Suppose first that

f is bounded, i.e. \lim _{t\to 0 ^{+}} f(t) = \alpha . A change of variable in the integral\int ^{inf ty} _{0} f(t)e ^{-st} dt shows that sF(s) = \int ^{inf ty} _{0} f\left( (ts\right)e)/(^{-t}) dt .Since f is bounded, the Dominated Convergence Theorem implies that\lim _{s\toinfty} sF(s) = \int ^{inf ty\alpha} _{0} e ^{-t} dt = \alpha.

Proof using elementary calculus and assuming that function is bounded

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 ^{inf ty} _{A} e ^{-t} dt<\epsilon , and thennote that \lim _{s\toinfty} f\left( (ts\right) = \alpha)/() uniformly for t\in( 0 ,A] .

Generalizing to non-bounded functions that have exponential order

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 inf ty} sG(s) .But f( 0 ^{+)=g(0} ^{+)} and G(s) = F(s + c) , so\lim _{s\toinfty} sF(s) = \lim _{s\toinfty} (s - c)F(s) = \lim _{s\toinfty} sF(s + c) = \lim _{s\toinfty} sG(s), since \lim _{s\toinfty} F(s) =0 .

See also

• Final value theorem

Notes

1. Book: Fourier and Laplace transforms . 2003 . Cambridge University Press . R. J. Beerends . 978-0-511-67510-2 . Cambridge . 593333940.
2. Robert H. Cannon, Dynamics of Physical Systems, Courier Dover Publications, 2003, page 567.