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Find power series if you know its laplace transformation

  1. May 17, 2014 #1
    1. The problem statement, all variables and given/known data
    a) Determine power series ##\sum _{n=0}^{\infty }a_nt^n## if you know that its laplace transformation is ##-s^{-1}e^{-s^{-1}}##
    b) Determine function ##g## that this power series will be equal to ##J_0(g(t))##


    2. Relevant equations



    3. The attempt at a solution

    Hmmm, I am having some troubles with this laplace transformation in part a).

    Well, I know that Laplace transformation of Heaviside function ##H_C(t)## is ##\frac{1}{s}e^{-Cs}##

    Knowing this I get almost the same as the problem says: ##-H_1## ---> ##-\frac{1}{s}e^{-s}##. But I have absolutely NO idea what to do to get ##\frac{1}{s}## in the exponent function. If I just power both sides of the equation everything else collapses...

    So how can I deal with this?
     
  2. jcsd
  3. May 17, 2014 #2

    vela

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    Expand the exponential as a series then invert the transform term by term.
     
  4. May 17, 2014 #3
    Hmmmm...

    ##-\frac{1}{s}e^{-1/s}=-\frac{1}{s}(1-\frac{1}{s}+\frac{1}{2}(\frac{1}{s})^2-\frac{1}{3}(\frac{1}{s})^3+\frac{1}{4}(\frac{1}{s})^4- ...)=-\frac{1}{s}+(\frac{1}{s})^2-\frac{1}{2}(\frac{1}{s})^3+\frac{1}{3}(\frac{1}{s})^4-\frac{1}{4}(\frac{1}{s})^5+ ...##

    Using inverse Laplace transformation gives me:

    ##-1+t-\frac{1}{4}t^2+\frac{1}{18}t^3-\frac{1}{96}t^4+\frac{1}{600}t^5-\frac{1}{4320}t^6+...##

    BUT I can't fine a way to include that -1 into series:

    ##-1+\sum _{k=1}^{\infty }\frac{(-1)^{k+1}}{(k+1)!-k!}t^k##
     
  5. May 17, 2014 #4

    vela

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    You didn't expand the exponential function correctly. It should be n! in the denominator, not n.
     
  6. May 17, 2014 #5
    uh, jup, you are right.

    The result is ##\sum _{k=0}^{\infty }\frac{(-1)^{k+1}}{(k!)^2}t^k##.

    Thank you, vela!
     
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