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Mathematics LibreTexts

8.8: A Brief Table of Laplace Transforms

  • Page ID
    9587
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    Table \( \PageIndex{1}\) 
    \( \displaystyle f(t)\) \( \displaystyle F(s)\)  
    1 \( \displaystyle 1\over s\) \( \displaystyle (s > 0)\)
    \( \displaystyle t^n\) \( \displaystyle n!\over s^{n+1}\) \( \displaystyle (s > 0)\)
    (\( \displaystyle n = \mbox{ integer } > 0\))    
    \( \displaystyle t^p,\; p > -1\) \( \displaystyle \Gamma (p+1) \over s^{(p+1)}\) \( \displaystyle (s>0)\)
    \( \displaystyle e^{at}\) \( \displaystyle 1 \over s-a\) \( \displaystyle (s > a)\)
    \( \displaystyle t^ne^{at}\) \( \displaystyle n! \over (s-a)^{n+1}\) \( \displaystyle (s > 0)\)
    (\( \displaystyle n= \text{ integer } > 0\))    
    \( \displaystyle \cos \omega t\) \( \displaystyle \frac{s}{s^{2}+\omega ^{2}}\) \( \displaystyle (s > 0)\)
    \( \displaystyle \sin \omega t\) \( \displaystyle \omega \over s^2+\omega^2\) \( \displaystyle (s > 0)\)
    \( \displaystyle e^{\lambda t} \cos \omega t\) \( \displaystyle s - \lambda \over (s-\lambda)^2+\omega^2\) \( \displaystyle (s > \lambda)\)
    \( \displaystyle e^{\lambda t} \sin \omega t\) \( \displaystyle \omega \over (s-\lambda)^2+\omega^2\) \( \displaystyle (s > \lambda)\)
    \( \displaystyle \cosh bt\) \( \displaystyle s \over s^2-b^2\) \( \displaystyle (s > |b|)\)
    \( \displaystyle \sinh bt\) \( \displaystyle b \over s^2-b^2\) \( \displaystyle (s > |b|)\)
    \( \displaystyle t \cos \omega t\) \( \displaystyle s^2-\omega^2 \over (s^2+\omega^2)^2\) \( \displaystyle (s>0)\)
    \( \displaystyle t \sin \omega t\) \( \displaystyle 2\omega s \over (s^2+\omega^2)^2\) \( \displaystyle (s>0)\)
    \( \displaystyle \sin \omega t -\omega t\cos \omega t\) \( \displaystyle 2\omega^3\over (s^2+\omega^2)^2\) \( \displaystyle (s>0)\)
    \( \displaystyle \omega t - \sin \omega t\) \( \displaystyle \omega^3 \over s^2(s^2+\omega^2)^2\) \( \displaystyle (s>0)\)
    \( \displaystyle \frac{1}{t}\sin\omega t\) \( \displaystyle \arctan \left({\omega \over s}\right)\) \( \displaystyle (s>0)\)
    \( \displaystyle e^{at}f(t)\) \( \displaystyle F(s-a)\)  
    \( \displaystyle t^kf(t)\) \( \displaystyle (-1)^{k}F^{(k)}(s)\)  
    \( \displaystyle f(\omega t)\) \( \displaystyle \frac{1}{\omega}F\left(\frac{s}{\omega } \right), \quad \omega >0\)  
    \( \displaystyle u(t-\tau)\) \( \displaystyle e^{-\tau s} \over s\) \( \displaystyle (s>0)\)
    \( \displaystyle u(t-\tau)f(t-\tau)\, (\tau > 0)\) \( \displaystyle e^{-\tau s}F(s)\)  
    \( \displaystyle \displaystyle {\int^t_o f(\tau)g(t-\tau)\, d\tau}\) \( \displaystyle F(s) \cdot G(s)\)  
    \( \displaystyle \delta(t-a)\) \( \displaystyle e^{-as}\) \( \displaystyle (s>0)\)