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
  1. ^ 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)
  2. ^ Robert H. Cannon, Dynamics of Physical Systems, Courier Dover Publications, 2003, page 567.