Table of Contents

Formulas and Properties of Laplace Transform

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Formulas of Laplace Transform

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.


Properties of Laplace Transform

In what follows, function \( f(t) \) is written in small letters and its corresponding transform in capital letters \( F(s) \)
  1. Linearity
          If \( g(t) = a f_1(t) + b f_2(t) \), then \( G(s) = a F_1(s) + b F_2(s) \) , \( a \) and \( b \) are constants.
  2. Shift in t
          If \( g(t) = f(t - \tau) u( t - \tau) \), then \( G(s) = e^{- s \tau} F(s) \) , \( \tau \ge 0 \)
  3. Multiplication by an exponential in \( t \) results in a shift in \( s \)
          If \( g(t) = e^{-at} f(t) \), then \( G(s) = F(s + a) \) , \( a \ge 0 \)
  4. Scaling in \( t \)
          If \( g(t) = f(k t) \), then \( G(s) = \dfrac{1}{k} F(\dfrac{s}{k}) \)
  5. Derivative of \( F(s) \) with respect to \( s \)
          If \( g(t) = t f(t) \), then \( G(s) = - \dfrac{d F(s)}{d s} \)
  6. Derivative of \( f(t) \) with respect to \( t \)
          If \( g(t) = \dfrac{df(t)}{dt} = f'(t)\), then \( G(s) = s F(s) - f(0) \)
  7. Second derivative of \( f(t) \) with respect to \( t \)
          If \( g(t) = \dfrac{df^2(t)}{dt^2} = f''(t)\), then \( G(s) = s^2 F(s) - s f(0) - f'(0) \)
  8. \( n \) th derivative of \( f(t) \) with respect to \( t \)
         If \( g(t) = \dfrac{df^n(t)}{dt^n} = f^{(n)}(t)\),
         then \( G(s) = s^n F(s) - s^{n-1} f(0) - s^{n-2} f'(0) - ... - s f^{(n-2)}(0) - f^{(n-1)}(0) \)
  9. Integral of \( f(t) \) with respect to \( t \)
         If \( \displaystyle g(t) = \int_0^t f(t') dt'\) , then \( G(s) = \dfrac{1}{s} F(s) \)
  10. Convolution integral
         If \( \displaystyle g(t) = \int_0^t f_1(t')f_2(t-t') dt'\), then \( G(s) = F_1(s) F_2(s) \)



More References and Links

Definition of Laplace Transform .
Handbook of Mathematical Functions Page 1020.
Heaviside step function
Dirac Delta Functions
Laplace Transforms Computations Examples with Solutions.
Engineering Mathematics with Examples and Solutions