Table of Laplace transforms - System Analysis and Control

Here is an updated table of common Laplace transforms, including those involving operations on functions as well as common functions themselves. This table includes the formulas derived in the sections on Laplace transforms and Inverse Laplace transforms.

Table 1:Table of Laplace transforms for (a) operations and (b) common functions

Time-domain $f(t)$	Laplace transform $F(s)=\Lap\{f(t)\}$
$a f(t) + b g(t)$	$aF(s)+bG(s)$
$\dot{f}(t)$	$\displaystyle sF(s)-f(0)$
$\ddot{f}(t)$	$\displaystyle s^2F(s)-s f(0)-\dot{f}(0)$
$f^{(n)}(t) = \frac{d^n}{dt^n} f(t)$	$s^nF(s)-\sum_{k=0}^{n-1} s^{n-1-k} f^{(k)}(0)$
$\displaystyle\int_0^t f(\tau)\, d\tau$	$\dfrac{1}{s}F(s)$
$(f*g)(t)=\int_0^t f(\tau) g(t-\tau)\, d\tau$	$F(s)G(s)$
$H(t-\tau) f(t-\tau)$	$e^{-s\tau} F(s)$

(a)

Time-domain $f(t)$	Laplace transform $F(s)=\Lap\{f(t)\}$
$\delta(t)$	$\displaystyle 1$
$H(t)\;$	$\displaystyle \frac{1}{s}$
$t$	$\dfrac{1}{s^2}$
$t^n$	$\dfrac{n!}{s^{n+1}}$
$e^{-at}$	$\dfrac{1}{s+a}$
$\dfrac{t^{n-1}}{(n-1)!}\, e^{-at}$	$\dfrac{1}{(s+a)^{n}}$
$\cos(\omega t)$	$\dfrac{s}{s^2+\omega^2}$
$\sin(\omega t)$	$\dfrac{\omega}{s^2+\omega^2}$
$e^{-at}\cos(\omega t)$	$\dfrac{s+a}{(s+a)^2+\omega^2}$
$e^{-at}\sin(\omega t)$	$\dfrac{\omega}{(s+a)^2+\omega^2}$

(b)