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
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]
Proofs edit
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
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
since .
See also edit
Notes edit
- ^ 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) - ^ Robert H. Cannon, Dynamics of Physical Systems, Courier Dover Publications, 2003, page 567.