Initial value theorem
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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 : F(s) = \int_0^\infty f(t) e^\,dt 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 f is bounded on (0,\infty) (or if just f(t)=O(e^)) and \lim_f(t) exists then the initial value theorem saysRobert 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.
: \lim_f(t)=\lim_.


Proofs


Proof using dominated convergence theorem and assuming that function is bounded

Suppose first that f is bounded, i.e. \lim_f(t)=\alpha. A change of variable in the integral \int_0^\infty f(t)e^\,dt shows that :sF(s)=\int_0^\infty f\left(\frac ts\right)e^\,dt. Since f 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 :\lim_sF(s)=\int_0^\infty\alpha e^\,dt=\alpha.


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 A so that \int_A^\infty e^\,dt<\epsilon, and then note that \lim_f\left(\frac ts\right)=\alpha ''uniformly'' for t\in(0,A].


Generalizing to non-bounded functions that have exponential order

The theorem assuming just that f(t)=O(e^) follows from the theorem for bounded f: Define g(t)=e^f(t). Then g is bounded, so we've shown that g(0^+)=\lim_sG(s). But f(0^+)=g(0^+) and G(s)=F(s+c), so :\lim_sF(s)=\lim_(s-c)F(s)=\lim_sF(s+c) =\lim_sG(s), since \lim_F(s)=0.


See also

* Final value theorem


Notes

Theorems in analysis {{mathanalysis-stub