Table of Selected Laplace Transforms
====================================

These are from the `Wikipedia page on the Laplace transform <https://en.wikipedia.org/wiki/Laplace_transform#Table_of_selected_Laplace_transform>`_


.. list-table:: **Laplace Transform Properties**
   :header-rows: 1
   :class: tablefullwidth

   * - Function
     - Time Domain
     - $s$-Domain
     - ROC

   * - Unit pulse
     - $\delta(t)$
     - $1$
     - $\mathbb C$

   * - Delayed pulse
     - $\delta(t-\tau)$
     - $e^{-\tau s}$
     - $Re(s)>0$

   * - Unit step
     - $u(t)$
     - $\frac{1}{s}$
     - $Re(s)>0$

   * - Ramp
     - $t u(t)$
     - $\frac{1}{s^2}$
     - $Re(s)>0$

   * - Exponential decay
     - $e^{-\alpha t} u(t)$
     - $\frac{1}{s+\alpha}$
     - $Re(s)>-\alpha$

   * - Sine
     - $\sin(\omega t) u(t)$
     - $\frac{\omega}{s^2+\omega^2}$
     - $Re(s)>0$

   * - Cosine
     - $\cos(\omega t) u(t)$
     - $\frac{s}{s^2+\omega^2}$
     - $Re(s)>0$ 

   * - Exponentially Decaying Sine
     - $e^{-at}\sin(\omega t) u(t)$
     - $\frac{\omega}{(s+a)^2+\omega^2}$
     - $Re(s)>0$

   * - Exponentially Decaying Cosine
     - $e^{-at}\cos(\omega t) u(t)$
     - $\frac{s+a}{(s+a)^2+\omega^2}$
     - $Re(s)>0$