4. Laplace Transform of Common Functions

Function \(\boldsymbol{f(t)}\) \(\boldsymbol{F(s)}\)
Constant \(a\) \(\dfrac{a}{s}\)
Ramp \(at\) \(\dfrac{a}{s^2}\)
\(t^n\) \(\dfrac{n!}{s^{n+1}}\)
Exponential \(e^{-at}\) \(\dfrac{1}{s+a}\)
\(e^{at}\) \(\dfrac{1}{s-a}\)
\(t^n e^{-at}\) \(\dfrac{n!}{(s+a)^{n+1}}\)
Function \(\boldsymbol{f(t)}\) \(\boldsymbol{F(s)}\)
Sinusoidal \(\sin(\omega t)\) \(\dfrac{\omega}{s^2+\omega^2}\)
\(\cos(\omega t)\) \(\dfrac{s}{s^2+\omega^2}\)
Sin with exponential \(e^{-at}\sin(\omega t)\) \(\dfrac{\omega}{(s+a)^2+\omega^2}\)
\(e^{-at}\cos(\omega t)\) \(\dfrac{s+a}{(s+a)^2+\omega^2}\)
Hyperbolic \(\sinh(\omega t)\) \(\dfrac{\omega}{s^2-\omega^2}\)
\(\cosh(\omega t)\) \(\dfrac{s}{s^2-\omega^2}\)
Function \(\boldsymbol{f(t)}\) \(\boldsymbol{F(s)}\)
Dead time \(f(t - t_o)\) \(e^{-st_o}F(s)\)
Impulse \(\delta(t)\) 1
Rectangular pulse \(f(t) = \left\{\begin{array}{ll} 0 & t<0 \\ A & 0 < t < T \\ 0 & t>T \end{array} \right.\) \(\frac{A}{s}\left(1-e^{-sT}\right)\)