Definition: If \( f(t) \) is a one sided function such that \( f(t) = 0 \) for \( t \lt 0 \) then the Laplace transform \( F(s) \) is defined by
\[ \mathscr{L}\{f(t)\} = F(s) = \int_{0}^{+\infty} f(t) e^{-st} dt \]
where \( s \) is allowed to be a complex number for which the improper integral above converges.
A more precise definition of the Laplace function to accommodate for functions such as \( \delta(t) \) is given by
\[ \mathscr{L}\{f(t)\} = F(s) = \int_{0{-}}^{+\infty} f(t) e^{-st} dt \]
Laplace transforms computations with examples and solutions are included.
Function | Transform |
|---|---|
| \( f(t) \) | \( F(s) \) |
| \( u(t) \) | \( \dfrac{1}{s} \) |
| \( t^n \) | \( \dfrac{n!}{s^{n+1}} \) |
| \( e^{-at} \) | \( \dfrac{1}{s+a} \) |
| \( t^n e^{-at} \) | \( \dfrac{n!}{(s+a)^{n+1}} \) |
| \( \sin \omega t \) | \( \dfrac{\omega}{s^2+\omega^2} \) |
| \( t \sin \omega t \) | \( \dfrac{2 \omega s}{(s^2+\omega^2)^2} \) |
| \( \cos \omega t \) | \( \dfrac{s}{s^2+\omega^2} \) |
| \( t \cos \omega t \) | \( \dfrac{s^2 - \omega^2}{(s^2+\omega^2)^2} \) |
| \( \sinh \omega t \) | \( \dfrac{\omega}{s^2 - \omega^2} \) |
| \( \cosh \omega t \) | \( \dfrac{s }{s^2 - \omega^2} \) |
| \( \delta( t - \tau) \) | \( e^{-s \tau} \) , \( \tau \ge 0 \) |
| \( u( t - \tau) \) | \( \dfrac{1}{s} e^{-s \tau} \) , \( \tau \ge 0 \) |
Note
1) \( \delta( t ) \) is the Dirac delta function also called impulse function in engineering.
2) \( u( t) \) is the Heaviside step function.