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Initial value theorem
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<h2 id="proofs">Proofs</h2> <h3>Proof using dominated convergence theorem and assuming that function is bounded</h3> <p>Suppose first that f {\displaystyle f} is bounded, i.e. lim t → 0 + f ( t ) = α {\displaystyle \lim _{t\to 0^{+}}f(t)=\alpha } . A change of variable in the integral ∫ 0 ∞ f ( t ) e − s t d t {\displaystyle \int _{0}^{\infty }f(t)e^{-st}\,dt} shows that </p> s F ( s ) = ∫ 0 ∞ f ( t s ) e − t d t {\displaystyle sF(s)=\int _{0}^{\infty }f\left({\frac {t}{s}}\right)e^{-t}\,dt} . <p>Since f {\displaystyle f} is bounded, the <a href="/facts/Dominated_Convergence_Theorem/EDCoCBJY">Dominated Convergence Theorem</a> implies that </p> lim s → ∞ s F ( s ) = ∫ 0 ∞ α e − t d t = α . {\displaystyle \lim _{s\to \infty }sF(s)=\int _{0}^{\infty }\alpha e^{-t}\,dt=\alpha .} <h3>Proof using elementary calculus and assuming that function is bounded</h3> <p>Of course we don't really need DCT here, one can give a very simple proof using only elementary calculus: </p><p>Start by choosing A {\displaystyle A} so that ∫ A ∞ e − t d t < ϵ {\displaystyle \int _{A}^{\infty }e^{-t}\,dt<\epsilon } , and then note that lim s → ∞ f ( t s ) = α {\displaystyle \lim _{s\to \infty }f\left({\frac {t}{s}}\right)=\alpha } <i>uniformly</i> for t ∈ ( 0 , A ] {\displaystyle t\in (0,A]} . </p> <h3>Generalizing to non-bounded functions that have exponential order</h3> <p>The theorem assuming just that f ( t ) = O ( e c t ) {\displaystyle f(t)=O(e^{ct})} follows from the theorem for bounded f {\displaystyle f} : </p><p>Define g ( t ) = e − c t f ( t ) {\displaystyle g(t)=e^{-ct}f(t)} . Then g {\displaystyle g} is bounded, so we've shown that g ( 0 + ) = lim s → ∞ s G ( s ) {\displaystyle g(0^{+})=\lim _{s\to \infty }sG(s)} . But f ( 0 + ) = g ( 0 + ) {\displaystyle f(0^{+})=g(0^{+})} and G ( s ) = F ( s + c ) {\displaystyle G(s)=F(s+c)} , so </p> lim s → ∞ s F ( s ) = lim s → ∞ ( s − c ) F ( s ) = lim s → ∞ s F ( s + c ) = lim s → ∞ s G ( s ) , {\displaystyle \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),} <p>since lim s → ∞ F ( s ) = 0 {\displaystyle \lim _{s\to \infty }F(s)=0} . </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Final_value_theorem/uQ2hdoUY">Final value theorem</a></li></ul> <h2 id="notes">Notes</h2> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Fourier and Laplace transforms. R. J. Beerends. Cambridge: Cambridge University Press. 2003. ISBN 978-0-511-67510-2. OCLC 593333940.{{cite book}}: CS1 maint: others (link) <a href="978-0-511-67510-2" target="_blank">978-0-511-67510-2</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Robert H. Cannon, Dynamics of Physical Systems, Courier Dover Publications, 2003, page 567. <a href="/wiki/Courier_Dover_Publications" target="_blank">/wiki/Courier_Dover_Publications</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>