# Initial value theorem

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{{Short description|Mathematical theorem using Laplace transform}}
In [mathematical analysis](/source/mathematical_analysis), the '''initial value theorem''' is a theorem used to relate [frequency domain](/source/frequency_domain) expressions to the [time domain](/source/time_domain) behavior as time approaches [zero](/source/zero).<ref>{{Cite book |title=Fourier and Laplace transforms |date=2003 |publisher=Cambridge University Press |others=R. J. Beerends |isbn=978-0-511-67510-2 |location=Cambridge |oclc=593333940}}</ref>

Let

: <math> F(s) = \int_0^\infty f(t) e^{-st}\,dt </math>

be the (one-sided) [Laplace transform](/source/Laplace_transform) of ''&fnof;''(''t'').  If <math>f</math> is bounded on <math>(0,\infty)</math> (or if just <math>f(t)=O(e^{ct})</math>) and <math>\lim_{t\to 0^+}f(t)</math> exists then the initial value theorem says<ref>Robert H. Cannon, ''Dynamics of Physical Systems'', [Courier Dover Publications](/source/Courier_Dover_Publications), 2003, page 567.</ref>

: <math>\lim_{t\,\to\, 0}f(t)=\lim_{s\to\infty}{sF(s)}. </math>

== Proofs ==

=== Proof using dominated convergence theorem and assuming that function is bounded ===
Suppose first that <math> f</math> is bounded, i.e. <math>\lim_{t\to 0^+}f(t)=\alpha</math>. A change of variable in the integral
<math>\int_0^\infty f(t)e^{-st}\,dt</math> shows that 
:<math>sF(s)=\int_0^\infty f\left(\frac ts\right)e^{-t}\,dt</math>.
Since <math>f</math> is bounded, the [Dominated Convergence Theorem](/source/Dominated_Convergence_Theorem) implies that
:<math>\lim_{s\to\infty}sF(s)=\int_0^\infty\alpha e^{-t}\,dt=\alpha.</math>

=== 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 <math>A</math> so that <math>\int_A^\infty e^{-t}\,dt<\epsilon</math>, and then
note that <math>\lim_{s\to\infty}f\left(\frac ts\right)=\alpha</math> ''uniformly'' for <math>t\in(0,A]</math>.

=== Generalizing to non-bounded functions that have exponential order ===
The theorem assuming just that <math>f(t)=O(e^{ct})</math> follows from the theorem for bounded <math>f</math>:

Define <math>g(t)=e^{-ct}f(t)</math>. Then <math>g</math> is bounded, so we've shown that <math>g(0^+)=\lim_{s\to\infty}sG(s)</math>.
But <math>f(0^+)=g(0^+)</math> and <math>G(s)=F(s+c)</math>, so
:<math>\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),</math>
since <math>\lim_{s\to\infty}F(s)=0</math>.

==See also==
* [Final value theorem](/source/Final_value_theorem)

==Notes==
<references/>

Category:Theorems in mathematical analysis

{{mathanalysis-stub}}

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Adapted from the Wikipedia article [Initial value theorem](https://en.wikipedia.org/wiki/Initial_value_theorem) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Initial_value_theorem?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.