Initial value theorem

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 {displaystyle F(s)=int _{0}^{infty }f(t)e^{-st},dt} be the (one-sided) Laplace transform of ƒ(t). If {displaystyle f} is bounded on {displaystyle (0,infty )} (or if just {displaystyle f(t)=O(e^{ct})} ) and {displaystyle lim _{tto 0^{+}}f(t)} exists then the initial value theorem says[2] {displaystyle lim _{t,to ,0}f(t)=lim _{sto infty }{sF(s)}.} Proof Suppose first that {displaystyle f} is bounded. Say {displaystyle lim _{tto 0^{+}}f(t)=alpha } . A change of variable in the integral {displaystyle int _{0}^{infty }f(t)e^{-st},dt} shows that {displaystyle sF(s)=int _{0}^{infty }fleft({frac {t}{s}}right)e^{-t},dt} .

Since {displaystyle f} is bounded, the Dominated Convergence Theorem shows that {displaystyle lim _{sto 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 {displaystyle A} so that {displaystyle int _{A}^{infty }e^{-t},dt

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