Common Laplace Transforms

Function Time domain $$f(t) = \mathcal{L}^{-1}\{F(s)\}$$ Laplace s-domain $$F(s) = \mathcal{L}\{f(t)\}$$ Region of convergence Reference
unit impulse $$\delta(t)$$ $$1$$ $$\text{all } s$$ inspection
delayed impulse $$\delta(t - \tau)$$ $$e^{-\tau s}$$ $$\text{all } s$$ time shift of
unit impulse
unit step $$u(t)$$ $$\frac{1}{s}$$ $$\text{Re}\{s\} > 0$$ integrate unit impulse
delayed unit step $$u(t - \tau)$$ $$\frac{e^{-\tau s}}{s}$$ $$\text{Re}\{s\} > 0$$ time shift of
unit step
ramp $$t \cdot u(t)$$ $$\frac{1}{s^2}$$ $$\text{Re}\{s\} > 0$$ integrate unit
impulse twice
nth power
(for integer $n$)		 $$\frac{t^n}{n!} \cdot u(t)$$ $$\frac{1}{s^{n+1}}$$ $\text{Re}\{s\} > 0$
$n \gt -1$		 Integrate unit
step $n$ times
nth power
with frequency shift		 $$\frac{t^n}{n!} \cdot e^{-\alpha t} \cdot u(t)$$ $$\frac{1}{(s + \alpha)^{n+1}}$$ $$\text{Re}\{s\} > -\alpha$$ Integrate unit step,
apply frequency shift
delayed nth power
with frequency shift		 $$\frac{(t - \tau)^n}{n!} \cdot e^{-\alpha(t - \tau)} \cdot u(t - \tau)$$ $$\frac{e^{-\tau s}}{(s + \alpha)^{n+1}}$$ $$\text{Re}\{s\} > -\alpha$$ Integrate unit step,
apply frequency shift,
apply time shift
qth power
(for complex $q$)		 $$\frac{t^q}{\Gamma(q + 1)} \cdot u(t)$$ $$\frac{1}{s^{q+1}}$$ $\text{Re}\{s\} > 0$
$\text{Re}\{q\} > -1 $		 Integrate unit step $n$ times,
apply time shift
exponential decay $$e^{-\alpha t} \cdot u(t)$$ $$\frac{1}{s + \alpha}$$ $$\text{Re}\{s\} > -\alpha$$ Frequency shift of
unit step
two-sided exponential decay $$e^{-\alpha \left| t \right| } $$ $$\frac{2\alpha}{\alpha^2 - s^2}$$ $$-\alpha < \text{Re}\{s\} < \alpha$$ Frequency shift of
unit step
exponential approach $$\left(1 - e^{-\alpha t}\right) \cdot u(t)$$ $$\frac{\alpha}{s(s + \alpha)}$$ $$\text{Re}\{s\} > 0$$ Unit step minus
exponential decay
sine $$\sin(\omega t) \cdot u(t)$$ $$\frac{\omega}{s^2 + \omega^2}$$ $$\text{Re}\{s\} > 0$$ pythagorean in s-domain
cosine $$\cos(\omega t) \cdot u(t)$$ $$\frac{s}{s^2 + \omega^2}$$ $$\text{Re}\{s\} > 0$$ pythagorean in s-domain
hyperbolic sine $$\sinh(\alpha t) \cdot u(t)$$ $$\frac{\alpha}{s^2 - \alpha^2}$$ $$\text{Re}\{s\} > |\alpha|$$ s-domain shifted 90
hyperbolic cosine $$\cosh(\alpha t) \cdot u(t)$$ $$\frac{s}{s^2 - \alpha^2}$$ $$\text{Re}\{s\} > |\alpha|$$ s-domain shifted 90
Exponentially-decaying
sine wave		 $$e^{-\alpha t} \sin(\omega t) \cdot u(t)$$ $$\frac{\omega}{(s + \alpha)^2 + \omega^2}$$ $$\text{Re}\{s\} > -\alpha$$ frequency shift
Exponentially-decaying
cosine wave		 $$e^{-\alpha t} \cos(\omega t) \cdot u(t)$$ $$\frac{s + \alpha}{(s + \alpha)^2 + \omega^2}$$ $$\text{Re}\{s\} > -\alpha$$ frequency shift
nth root $$\sqrt[n]{t} \cdot u(t)$$ $$\frac{1}{s^{(n+1)/n}} \cdot \Gamma\left(1 + \frac{1}{n}\right)$$ $$\text{Re}\{s\} > 0$$ Domain extension
of factorial/gamma
natural logarithm $$\ln\left(\frac{t}{t_0}\right) \cdot u(t)$$ $$-\frac{t_0}{s} \left[\ln(t_0 s) + \gamma\right]$$ $$\text{Re}\{s\} > 0$$ inverse?
Bessel function
of the first kind,
of order $n$		 $$J_n(\omega t) \cdot u(t)$$ $$\frac{\left(\sqrt{s^2 + \omega^2} - s\right)^n}{\omega^n \sqrt{s^2 + \omega^2}}$$ $\text{Re}\{s\} > 0$
$ n > -1 $		 It's'a J (Bessel)
Error function $$\text{erf}(t) \cdot u(t)$$ $$\frac{e^{s^2/4} \left(1 - \text{erf}(s/2)\right)}{s}$$ $$\text{Re}\{s\} > 0$$ erf is an anagram for ref

Note: This table provides common Laplace transform pairs. The region of convergence (ROC) is crucial for the uniqueness of the inverse transform. For causal systems, the ROC is typically to the right of the rightmost pole. The notation $u(t)$ represents the unit step function.