Laplace transform

In mathematics and in particular, in functional analysis, the Laplace transform of a function f(t) defined for all real numbers t ≥ 0 is the function F(s), defined by:

mathcal{L}left{ 1 * f(t) ight} {1 over s} mathcal{L}{f}

• Initial value theorem

: f(0^+)=lim_{s o infty}{sF(s)}

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• Final value theorem

: f(infty)=lim_{s o 0}{sF(s)}, all poles in left-hand plane.

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: The final value theorem is useful because it gives the long-term behaviour without having to perform partial fraction decompositions or other difficult algebra. If a functions poles are in the right hand plane (e.g. e^t or sin(t)) the behaviour of this formula is undefined.

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• $s$ shifting

: mathcal{L}left{ e^{at} f(t) ight}

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