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Relation of laplace transform with power series

  1. Apr 12, 2012 #1
    Hi, just wonder if anyone can help

    1. The problem statement, all variables and given/known data

    Apparently there is a relation between laplace transform and power series. http://www.jstor.org/stable/pdfplus/2305640.pdf?acceptTC=true states that if the discrete variable n of a power series is replaced by a continuous variable lambda, the infinite sum becomes an integral with boundaries between zero and infinity. But where does the d'lambda' of the integrand come from? This looks a bit like the Riemann sum, but if I haven't understood it wrong, the dx in the Riemann integrand is supposed to be a replacement for the discrete delta x in the Riemann sum and thus the operation is said to be integrate with respect to x? There isn't any delta lambda in the original power series which the laplace transform is made analogous to?

    Second, I don't really know why one can arbitrarily plug in any formula into the laplace transform integral and get the transform of it. The whole procedure is just like evaluating a power series with the function of interest acting as the accompanying coefficients of a continuous power series infinite sum of [f(t)x^t], isn't that quite arbitrary? or did this manipulation just happen to turn out to be something quite useful in solving differential equations?

    Thanks.


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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