# Using the Laplace transform to evaluate the steady-state of a function

My understanding is that the Laplace transform evaluated at $s = i \omega t$ can be used to evaluate the steady-state of a function. How is this done? I can't find any information on this in my textbooks nor on the internet.

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possible duplicate of Frequency Response of Circuits - Laplace Transforms –  Fabian Mar 22 '12 at 21:33
I still believe the question might be better off in the physics or electrical engeneering stack exchange. –  Fabian Mar 22 '12 at 21:33
It just becomes a Fourier transform. –  Raskolnikov Mar 22 '12 at 21:34
A 1965 book is Murry Spiegel, Shaum's Outline of Theory and Problems of Laplace Transforms, McGraw-Hill. –  Américo Tavares Mar 22 '12 at 22:36
The Laplace transform can be used to evaluate the transient response of an electrical circuit when an input voltage is applied to it; or the behavior of a mechanical beam when is subjected to a load applied to it. Both situations can be modeled by differential equations, depending on the initial conditions. These are solved using the Laplace transform and afterwards the inverse transform is used to find the result. –  Américo Tavares Mar 22 '12 at 22:39
Let $f(t)$ denote the time-domain function, and $F(s)$ denote its Laplace transform. The final value theorem states that: $$\lim_{t \to \infty} f(t) = \lim_{s\to 0} sF(s),$$ where the LHS is the steady state of $f(t).$ Since it is typically hard to solve for $f(t)$ directly, it is much easier to study the RHS where, for example, ODEs become polynomials or rational functions in $s.$