In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, the initial value theorem is a theorem used to relate
frequency domain
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...
expressions to the
time domain
Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the c ...
behavior as time approaches
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
.
Let
:
be the (one-sided)
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
of ''ƒ''(''t''). If
is bounded on
(or if just
) and
exists then the initial value theorem says
[Robert H. Cannon, ''Dynamics of Physical Systems'', ]Courier Dover Publications
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, books ...
, 2003, page 567.
:
Proofs
Proof using dominated convergence theorem and assuming that function is bounded
Suppose first that
is bounded, i.e.
. A change of variable in the integral
shows that
:
.
Since
is bounded, the
Dominated Convergence Theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary t ...
implies that
:
Proof using elementary calculus and assuming that function is bounded
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
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
*
Final value theorem
Notes
Theorems in analysis
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