Initial Value Theorem
From ControlTheoryPro.com

Contents 
1 Introduction to Initial Value Theorem
In mathematics, the initial value theorem is a theorem used to relate frequency domain expressions to the time domain behavior as time approaches zero.^{[1]}
The Laplace Transform of is
Definition 
where
Definition 
be the (onesided) Laplace transform of . The initial value theorem then states^{[2]}
2 Continuous Time form of the Initial Value Theorem
Continuous Time 
2.1 Franklin et all's version^{[3]}
The Initial Value Theorem states that it is always possible to determine the initial vlaue of the time function from its Laplace Transform. Mathematically this can be stated as:
Eqn. 3.28 
2.1.1 Proof
Eqn. 3.29 
Consider when and rewrite as
Taking the limit of Eqn. 3.29 as , we get
The 2nd term on the right side approaches 0 because . So
or
2.1.2 Example
Find the initial value of the signal
Answer:
3 Discrete Time form of the Initial Value Theorem
Discrete Time 
4 References
 http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node17.html
 Robert H. Cannon, Dynamics of Physical Systems, Courier Dover Publications, 2003
 Franklin, G. F., EmamiNaeini, A., and Powell, J. D. 1993 Feedback Control of Dynamic Systems. 3rd. AddisonWesley Longman Publishing Co., Inc. ISBN 0201527472
4.1 Notes
 ↑ http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node17.html, 4/3/09
 ↑ Cannon, pg 567
 ↑ Franklin et all, pg 105