1 |
\( \displaystyle{ 1\over s}\) |
\( \displaystyle (s > 0)\) |
\( \displaystyle t^n\) |
\( \displaystyle{ n!\over s^{n+1} }\) |
\( \displaystyle (s > 0)\) |
(\( \displaystyle n = \mbox{ integer } > 0\)) |
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\( \displaystyle t^p,\; p > -1\) |
\( \displaystyle{ \Gamma (p+1) \over s^{(p+1)} }\) |
\( \displaystyle (s>0)\) |
\( \displaystyle e^{at}\) |
\( \displaystyle{ 1 \over s-a }\) |
\( \displaystyle (s > a)\) |
\( \displaystyle t^ne^{at}\) |
\( \displaystyle{ n! \over (s-a)^{n+1} }\) |
\( \displaystyle (s > 0)\) |
(\( \displaystyle n= \text{ integer } > 0\)) |
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\( \displaystyle \cos \omega t\) |
\( \displaystyle{ \frac{s}{s^{2}+\omega ^{2}} }\) |
\( \displaystyle (s > 0)\) |
\( \displaystyle \sin \omega t\) |
\({ \displaystyle \omega \over s^2+\omega^2 }\) |
\( \displaystyle (s > 0)\) |
\( \displaystyle e^{\lambda t} \cos \omega t\) |
\( \displaystyle{ s - \lambda \over (s-\lambda)^2+\omega^2 }\) |
\( \displaystyle (s > \lambda)\) |
\( \displaystyle e^{\lambda t} \sin \omega t\) |
\( \displaystyle{ \omega \over (s-\lambda)^2+\omega^2 }\) |
\( \displaystyle (s > \lambda)\) |
\( \displaystyle \cosh bt\) |
\( \displaystyle{ s \over s^2-b^2 }\) |
\( \displaystyle (s > |b|)\) |
\( \displaystyle \sinh bt\) |
\( \displaystyle{ b \over s^2-b^2 }\) |
\( \displaystyle (s > |b|)\) |
\( \displaystyle t \cos \omega t\) |
\( \displaystyle{ s^2-\omega^2 \over (s^2+\omega^2)^2 }\) |
\( \displaystyle (s>0)\) |
\( \displaystyle t \sin \omega t\) |
\( \displaystyle{ 2\omega s \over (s^2+\omega^2)^2 }\) |
\( \displaystyle (s>0)\) |
\( \displaystyle \sin \omega t -\omega t\cos \omega t\) |
\( \displaystyle{ 2\omega^3\over (s^2+\omega^2)^2 }\) |
\( \displaystyle (s>0)\) |
\( \displaystyle \omega t - \sin \omega t\) |
\( \displaystyle{ \omega^3 \over s^2(s^2+\omega^2) }\) |
\( \displaystyle (s>0)\) |
\( \displaystyle \frac{1}{t}\sin\omega t\) |
\( \displaystyle{ \arctan \left({\omega \over s}\right) }\) |
\( \displaystyle (s>0)\) |
\( \displaystyle e^{at}f(t)\) |
\( \displaystyle{ F(s-a) }\) |
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\( \displaystyle t^kf(t)\) |
\( \displaystyle (-1)^{k}F^{(k)}(s)\) |
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\( \displaystyle f(\omega t)\) |
\( \displaystyle{ \frac{1}{\omega}F\left(\frac{s}{\omega } \right), \quad \omega >0 }\) |
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\( \displaystyle u(t-\tau)\) |
\( \displaystyle{ e^{-\tau s} \over s }\) |
\( \displaystyle (s>0)\) |
\( \displaystyle u(t-\tau)f(t-\tau)\, (\tau > 0)\) |
\( \displaystyle{ e^{-\tau s}F(s) }\) |
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\( \displaystyle \displaystyle {\int^t_o f(\tau)g(t-\tau)\, d\tau}\) |
\( \displaystyle{ F(s) \cdot G(s) }\) |
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\( \displaystyle \delta(t-a)\) |
\( \displaystyle{ e^{-as} }\) |
\( \displaystyle (s>0)\) |