Tabela transformat Laplace’a
\( F(z)= {\mathcal L}(f(t))(z)=\displaystyle\int_0^{+\infty}e^{-zt}f(t)dt. \)
Tabela 1: Transformaty Laplace'a
| \( \hskip 0.6pc f(t)\hskip 0.6pc \) | \( \hskip 0.6pc F(z)= {\mathcal L}(f(t)(z)\hskip 0.6pc \) | \( \hskip 0.6pc f(t)\hskip 0.6pc \) | \( \hskip 0.6pc F(z)= {\mathcal L}(f(t)(z)\hskip 0.6pc \) |
| \( \hskip 0.6pc 1\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac {1^{}}z\hskip 0.6pc \) | \( \hskip 0.6pc e^{\lambda t}\sin\omega t\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac{\omega}{(z-\lambda )^2+\omega ^2}\hskip 0.6pc \) |
| \( \hskip 0.6pc t^n\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac{n!}{z^{n+1}}\hskip 0.6pc \) | \( \hskip 0.6pc e^{\lambda t}\cos\omega t\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac{z-\lambda }{(z-\lambda )^2+\omega ^2}\hskip 0.6pc \) |
| \( \hskip 0.6pc \sin t\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac{1}{z^2+1}\hskip 0.6pc \) | \( \hskip 0.6pc {\rm sinh}\,\omega t\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac{1 }{z^2-\omega ^2}\hskip 0.6pc \) |
| \( \hskip 0.6pc \cos t\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac{z}{z^2+1}\hskip 0.6pc \) | \( \hskip 0.6pc {\rm cosh}\,\omega t\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac{z }{z^2-\omega ^2}\hskip 0.6pc \) |
| \( \hskip 0.6pc e^{\lambda t}\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac{1}{z-\lambda }\hskip 0.6pc \) | \( \hskip 0.6pc e^{\lambda t}{\rm sinh}\,\omega t\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac{\omega }{(z-\lambda )^2-\omega ^2}\hskip 0.6pc \) |
| \( \hskip 0.6pc e^{\lambda t} t^n\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac{n!}{(z-\lambda )^{n+1}}\hskip 0.6pc \) | \( \hskip 0.6pc e^{\lambda t}{\rm cosh}\,\omega t\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac{z }{(z-\lambda )^2-\omega ^2}\hskip 0.6pc \) |
| \( \hskip 0.6pc \sin \omega t\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac{\omega}{z^2+\omega^2}\hskip 0.6pc \) | \( \hskip 0.6pc \delta(t)\hskip 0.6pc \) | \( \hskip 0.6pc 1\hskip 0.6pc \) |
| \( \hskip 0.6pc \cos \omega t\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac{z}{z^2+\omega^2}\hskip 0.6pc \) | \( \hskip 0.6pc \delta(t-\alpha )\hskip 0.6pc \) | \( \hskip 0.6pc e^{-\alpha z}\hskip 0.6pc \) |
| \( \hskip 0.6pc t^{\alpha}\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac{\Gamma (\alpha +1) }{z^{\alpha +1}}\hskip 0.6pc \) | \( \hskip 0.6pc e^{\lambda t}t^{\alpha}\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac{\Gamma (\alpha +1) }{(z-\lambda )^{\alpha +1}}\hskip 0.6pc \) |
| \( \hskip 0.6pc H(t-\alpha )\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac {e^{-\alpha z}}z\hskip 0.6pc \) | \( \hskip 0.6pc e^{\lambda (t-\alpha )}H(t-\alpha )\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac {e^{-\alpha z}}{z-\lambda }\hskip 0.6pc \) |
| \( \hskip 0.6pc \dfrac{\sin \omega t}t\hskip 0.6pc \) | \( \hskip 0.6pc {\rm arc tg}\dfrac{\omega }z\hskip 0.6pc \) | \( \hskip 0.6pc (t-\alpha )^n H(t-\alpha )\hskip 0.6pc \) | \( \hskip 0.6pc e^{-\alpha z}\dfrac{n!}{z^{n+1} }\hskip 0.6pc \) |
| \( \hskip 0.6pc t\sin \omega t\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac{2\omega z}{(z^2+\omega^2)^2}\hskip 0.6pc \) | \( \hskip 0.6pc (1-\alpha t)e^{-\alpha t}\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac z{(z+\alpha )^2}\hskip 0.6pc \) |
| \( \hskip 0.6pc t\cos \omega t\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac{z^2-\omega^2}{(z^2+\omega^2)^2}\hskip 0.6pc \) | \( \hskip 0.6pc \big(t-\dfrac{\alpha t^2}2\big)e^{-\alpha t}\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac z{(z+\alpha )^3}\hskip 0.6pc \) |
| \( \hskip 0.6pc \sin^2 \omega t\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac{2\omega^2}{z(z^2+4\omega^2)}\hskip 0.6pc \) | \( \hskip 0.6pc (1-e^{-t})^n\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac{n!}{z(z+1)\ldots (z+n)}\hskip 0.6pc \) |
| \( \hskip 0.6pc \cos^2 \omega t\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac{z^2+2\omega^2}{z(z^2+4\omega^2)}\hskip 0.6pc \) | \( \hskip 0.6pc (1-\alpha t)e^{-\alpha t}\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac z{(z+\alpha )^2}\hskip 0.6pc \) |
| \( \hskip 0.6pc \dfrac{1-\cos \omega t}{\omega^2 }\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac 1{z(z^2+\omega^2)}\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac{\sin \omega t-\omega t\cos \omega t}{2\omega^2 }\hskip 0.6pc \) | \( \hskip 0.6pc \dfrac 1{(z^2+\omega^2)^2}\hskip 0.6pc \) |