Differentiation
The differentiation theorem for Laplace transforms states that
![$\displaystyle {\dot x}(t) \leftrightarrow s X(s) - x(0),
$](http://www.dsprelated.com/josimages_new/filters/img1725.png)
![$ {\dot x}(t) \isdef \frac{d}{dt}x(t)$](http://www.dsprelated.com/josimages_new/filters/img1726.png)
![$ x(t)$](http://www.dsprelated.com/josimages_new/filters/img1659.png)
![$ t$](http://www.dsprelated.com/josimages_new/filters/img185.png)
![$\displaystyle \zbox {{\cal L}_{s}\{{\dot x}\} = s X(s) - x(0).}
$](http://www.dsprelated.com/josimages_new/filters/img1727.png)
Proof:
This follows immediately from integration by parts:
![\begin{eqnarray*}
{\cal L}_{s}\{{\dot x}\} &\isdef & \int_{0}^\infty {\dot x}(t)...
...y} -
\int_{0}^\infty x(t) (-s)e^{-s t} dt\\
&=& s X(s) - x(0)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/filters/img1728.png)
since
by assumption.
Corollary: Integration Theorem
![$\displaystyle \zbox {{\cal L}_{s}\left\{\int_0^t x(\tau)d\tau\right\} = \frac{X(s)}{s}}
$](http://www.dsprelated.com/josimages_new/filters/img1730.png)
Thus, successive time derivatives correspond to successively higher
powers of , and successive integrals with respect to time
correspond to successively higher powers of
.
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