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Laplace Transform Question

  1. Nov 13, 2005 #1
    I need to show that for f(t)=f(t+T) on [0,infty), that the Laplace Transform is:

    [tex]\mathcal{L}\left[f(t)\right]=\frac{\int_0^Te^{-st}f(t)\,dt}{1-e^{-sT}}.[/tex]

    The first thing I did was to write the transform as:

    [tex]\mathcal{L}\left[f(t)\right]=\sum_{n=0}^{\infty}\int_{nT}^{\left(n+1\right)T}e^{-st}f(t)\,dt.[/tex]

    Am I on the right track here? It looks like the formula given to me (that I need to show) is an infinite geometric series multiplied by the integral in the numerator. However, I am unable to get what I have into something of that form. Any ideas?

    Thank you.
     
  2. jcsd
  3. Nov 13, 2005 #2

    Physics Monkey

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    Yes, you are practically done already. Make a good change of variables, use the periodicity of f, and you're home free.
     
  4. Nov 13, 2005 #3
    Could you please elaborate on that a bit more? Thank you.
     
  5. Nov 13, 2005 #4

    Physics Monkey

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    Sure, you want each term to give [tex] \int^T_0 e^{-st} f(t) dt [/tex] times the geometric series part, right? So why not try to make a change of variable in each term to see if you can get this out? Make the limits look right for each term and see where that leads you.
     
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