1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Laplace transform of derivative of convolution

  1. Apr 22, 2017 #1
    Prelude
    Consider the convolution h(t) of two function f(t) and g(t):
    $$h(t) = f(t) \ast g(t)=\int_0^t f(t-\tau) g(\tau) d \tau$$
    then we know that by the properties of convolution
    $$\frac{d h(t)}{d t} = \frac{d f(t)}{d t} \ast g(t) = f(t) \ast \frac{d g(t)}{d t}$$

    Intermezzo
    We also know that the Laplace Transform of the convolution is given :
    $$H(s) = F(s) G(s)$$
    and that of the derivative of for example f(t) is
    $$s F(s) - f(0)$$

    Finale
    Therefore I should have for the Laplace transform of the derivative of the convolution:
    $$s H(s) = s F(s) G(s) - f(0) G(s) = F(s) s G(s) - F(s) g(0)$$

    But in general we don't have :
    $$ f(0) G(s) = F(s) g(0) $$

    Are this formulas correct? What is going on? Are they equivalent?
     
  2. jcsd
  3. Apr 22, 2017 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Your formula ##d h(t)/dt = (f'*g)(t)## is incorrect. Instead of trying to (mis-)apply canned formulas, just start with the definition ##h(t) = (f*g)(t) = \int_0^t f(t-\tau) g(\tau) \, d\tau## and differentiate it using standard results in calculus.
     
  4. Apr 22, 2017 #3
    Gotcha!

    $$\frac{d h(t)}{d t} =\frac{d}{ d t}\int_0^t f(t-\tau) g(\tau) d \tau $$
    and using Leibniz integral rule
    $$ \frac{d h(t)}{d t} = f(0)g(t) + \int_0^t \frac{d f(t-\tau)}{ d t} g(\tau) d \tau $$

    The laplace transform of wich is
    $$ s H(s) = G(s) f(0) + G(s)(s F(s) - f(0) ) = s F(s)G(s) $$

    Thank you!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Laplace transform of derivative of convolution
Loading...