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Limit Problem

  1. Mar 20, 2008 #1
    Say a function f and its derivative are everywhere continuous and of exponential order at infinity. F(s) is the Laplace transform of f(x). I need to find the limit of F as s goes to infinity.

    I use the integral definition of the Laplace transform and the fact that f is of exponential order. My problem is that I don't know if you can move the limit inside the integral. If you can, then it is clear that the result is 0. How can I justify this step, or is there a better approach?
     
  2. jcsd
  3. Mar 21, 2008 #2

    gel

    User Avatar

    You can use the dominated convergence theorem. It's stated in that link for general measure spaces, but you can just replace [itex]d\mu[/itex] with dx for integrating over the reals.

    or,

    if f is bounded by exp(ax), then bound f(x)exp(-sx) by exp(-(s-a)x). Use this to bound F(s) and you can calculate the rate at which it goes to 0.
     
    Last edited: Mar 21, 2008
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