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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
![{\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EzjBDoqdFaNo2oDC0nqsPnqzEythBnqsNaqw1otKQyqo4oDlEnqhF)
be the (one-sided) Laplace transform of ƒ(t). If
is bounded on
(or if just
) and
exists then the initial value theorem says[2]
![{\displaystyle \lim _{t\,\to \,0}f(t)=\lim _{s\to \infty }{sF(s)}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Caqi5ngsOoAzFaNCNztzFoDo3zga4nqdEnga3ajnEaqzCotC2zjJF)
Proof using dominated convergence theorem and assuming that function is bounded[edit]
Suppose first that
is bounded, i.e.
. A change of variable in the integral
shows that
.
Since
is bounded, the Dominated Convergence Theorem implies that
![{\displaystyle \lim _{s\to \infty }sF(s)=\int _{0}^{\infty }\alpha e^{-t}\,dt=\alpha .}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83ngnDngdBntoOzghEaDrBaNo5ztvBota1zjG1oqs3oNmNaqwQa2vB)
Proof using elementary calculus and assuming that function is bounded[edit]
Of course we don't really need DCT here, one can give a very simple proof using only elementary calculus:
Start by choosing
so that
, and then
note that
uniformly for
.
Generalizing to non-bounded functions that have exponential order[edit]
The theorem assuming just that
follows from the theorem for bounded
:
Define
. Then
is bounded, so we've shown that
.
But
and
, so
![{\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),}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FoDo3aNnEnAs5ajBFyjGNzjKOntsNzDa3oDi3ygsPngiOygwQnjK2)
since
.
See also[edit]