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Asymptotic expansion

  1. Oct 26, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the first two terms in an asymptotic expansion of the following as x goes to zero from the right (i.e. takes on smaller and smaller positive values).

    [itex]\int^{1}_{0}[/itex]e[itex]^{-x/t}[/itex]dt

    2. Relevant equations



    3. The attempt at a solution
    I'm not exactly sure how to proceed with this. I'm assuming I should expand the integrand as a Taylor series:

    e[itex]^{-x/t}[/itex]=[itex]\sum^{\infty}_{k=0}[/itex][itex]\frac{(-x/t)^{k}}{k!}[/itex]

    I'm kinda stuck here, though.
     
  2. jcsd
  3. Oct 26, 2012 #2

    Zondrina

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    Homework Helper

    Notice you start having problems as t goes to zero... you should re-write your integral with a limit and then I think it would be best to actually expand the terms of your sum this time.
     
  4. Oct 26, 2012 #3
    lim[itex]_{a\rightarrow0^{+}}[/itex][itex]\int^{a}_{0}[/itex](1-[itex]\frac{x}{t}[/itex]+[itex]\frac{x^{2}}{2t^{2}}[/itex]-[itex]\frac{x^{3}}{6t^{3}}[/itex]+[itex]\cdots[/itex])dt

    Like this? Hmm...
     
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