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Definition and properties Laplace

$$ F(s)=\int_0^\infty f(t) e^{-st}dt $$

$$ f(t)={1 \over 2\pi i}\int_{c-i\infty}^{c+i\infty} e^{st} F(s)ds $$

Residue Theorem: If $f(z)$ is analytic within and on $C$ except for a finite number of poles $$ \int_C f(z) dz = 2\pi i S $$ Where $S$ is the sum of the residues at the poles within $C$

Residue $$ a_{-1}=\lim_{ z\to a }{1 \over (n-1)!} {d^{n-1}\over dz^{n-1}}\bigl((z-a)^n f(z)\bigr)$$

Final value Theorem: $$ f(\infty)=\lim_{s\to 0} sF(s) $$ Initial value Theorem: $$ f(0^+)=\lim_{s\to \infty} sF(s) $$

A table of Laplace transforms

$\displaystyle f(t)$ $\displaystyle F(s)$
unit impulse $\displaystyle\delta(t)$ 1
unit impulse $\displaystyle\delta(t-t_0)$$\displaystyle e^{-st_0}$
Unit step u(t) $\displaystyle{1\over s}$
$u(t_0-t)$ $\displaystyle{{e^{st_0}\over s}}$
ramp of $kt$ $\displaystyle{{k\over s^2}}$
${df(t)\over dt}$ $sF(s)-f(0)$
$e^{-at}$ ${1 \over s+a}$
$e^{-t/T}$ ${T \over sT+1}$
$f(at)$ $ {1\over a}F({s\over a})$
$f*g$ $ FG$
$A\sin\omega t$ $ {A\omega\over s^2+\omega^2}$
$A\cos\omega t$ $ {As\over s^2+\omega^2}$
$\displaystyle \frac{{e}^{-a\,t}}{b-a}-\frac{{e}^{-b\,t}}{b-a} $ $\displaystyle \frac{1}{\left( a+s\right) \,\left( b+s\right) }$
$\displaystyle 1-{e}^{-a\,t}$ $\displaystyle \frac{a}{s\,\left( a+s\right) }$
Cosine arch $\displaystyle 1-\cos(2\pi t/T)$$\displaystyle \frac{1}{s}- \frac{e^{-sT}}{s} -\frac{s }{ s^2+\omega^2} + \frac{s e^{-sT} }{ s^2+\omega^2}$
sinc function $\displaystyle \frac{\mathrm{sin}\left( t\right) }{t}$$\displaystyle \frac\pi2-\mathrm{atan}\left( s\right)$
$\displaystyle \mathrm{e}^{-\beta\,t}\,\frac{1}{\sqrt{\beta^2-k}} \sinh\left(t\,\sqrt{\beta^2-k}\right)$$\displaystyle \frac1{(s^2+2\beta s+k)}$
$\displaystyle \frac{1}{k}\left(1-{\mathrm{e}}^{-\beta\,t}\,\left(\cosh\left(t\,\sqrt{\beta^2-k}\right)+\frac{\beta}{\sqrt{\beta^2-k}} \sinh\left(t\,\sqrt{\beta^2-k}\right) \right)\right)$$\displaystyle \frac1{s(s^2+2\beta s+k)}$
$\displaystyle \frac{{e}^{-\omega\,t\,\zeta}\,\mathrm{sin}\left( \omega\,t\,\sqrt{1-{\zeta}^{2}}\right) }{\sqrt{1-{\zeta}^{2}}}$$\displaystyle \frac{\omega}{{\omega}^{2}+2\,\zeta\,\omega\,s+{s}^{2}} $
$\displaystyle 1- {e}^{-\omega\,t\,\zeta}\,\left(\frac{\zeta\,\mathrm{sin}\left( \omega\,t\,\sqrt{1-{\zeta}^{2}}\right) }{\sqrt{1-{\zeta}^{2}}}+\mathrm{cos}\left( \omega\,t\,\sqrt{1-{\zeta}^{2}}\right) \right)$$\displaystyle \frac{{\omega}^{2}}{s\,\left( {\omega}^{2}+2\,\zeta\,\omega\,s+{s}^{2}\right) }$

For the second order systems $\omega$ is the natural frequency $\omega=\sqrt{k/m}$ and $\zeta$ is the damping where $2\zeta\omega={b/m}$. $2\beta=2\zeta\omega=b$

Laplace transform properties

Laplace of a differential

if $x(t)$ then Laplace transform of $\frac{dx}{dt}$ is \[ L(\frac{dx}{dt})=\int_0^\infty \frac{dx}{dt} e^{-st}dt=sx(s)+x(0) \]

By recursion it is then possible to compute the Laplace transform of $\frac{d^2x}{dt^2}$

z-transform

$$ F(z)=\sum_{n=-\infty}^\infty f[n] z^{-n} $$ where $z=e^{j\omega t}$

Shifting Theorem. if $f[n] \leftrightarrow F(z)$

then $f[n-m] \leftrightarrow z^{-m}F(z)$

Convolution Theorem. if $f_1[n] \leftrightarrow F_1(z)$ and $f_2[n] \leftrightarrow F_2(z)$ then $f_1 \otimes f_2 \leftrightarrow F_1F_2$

Final value Theorem: $$ f[\infty]=\lim_{z\to 1} (z-1)F(z) $$ Initial value Theorem: $$ f[0]=\lim_{z\to \infty} F(z) $$

Mapping from S-plane to Z-plane to W-plane