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Evaluating the function

  1. Sep 24, 2003 #1
    Let be the function

    f(t)=L(-1)1/(exp(-s)-1) where L(-1) means the laplace inverse transform..my doubt is to know what is the value of

    f(0),f(1),f(2)...f(n) being n an integer.
  2. jcsd
  3. Sep 24, 2003 #2

    Tom Mattson

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    I'll take a stab at this.

    First, we need to develop the Laplace transform of a periodic function:

    f(t)=f(t+nT) for all integers n.

    Start with the definition of L{f(t)}:


    Let's break this up over intervals of width T:


    Now perform substitutions on each integral such that the limits of each integral are [0,T]:


    Noting that f(t)=f(t+T)=f(t+2T)=f(t+3T)=..., and factoring ∫0Te-stf(t)dt out of each factor yields:


    The first factor on the right is a geometric series whose sum is:


    So, I have finally:


    Now, we get your function if we let f(t)=d(t+n+1/2). That is, a periodic delta function whose period is 1. I used the half integer n+1/2 so that there is only one delta function in each interval. I could just as easily have chosen n+1/3, n+1/4, or whatever. So, it seems that the function is not unique.

    Does that help?

    edit: fixed a variety of bracket errors
    Last edited: Sep 24, 2003
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