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Laplace's Method(Integration)

  1. Apr 18, 2014 #1

    wel

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    Gold Member

    Consider the integral
    \begin{equation}
    I(x)=\int^{2}_{0} (1+t) \exp\left(x\cos\left(\frac{\pi(t-1)}{2}\right)\right) dt
    \end{equation}
    Use Laplace's Method to show that
    \begin{equation}
    I(x) \sim \frac{4\sqrt{2}e^{x}}{\sqrt{\pi x}} \end{equation}
    as [itex]x\rightarrow\infty[/itex].

    => I have tried using the expansion of [itex]I(x)[/itex] in McLaurin series but did not get the answer.
    here,
    \begin{equation}
    h(t)=cos(\frac{\pi(t-1)}{2})
    \end{equation}
    [itex]h(0)= 0[/itex]

    [itex]h'(0)= \frac {\pi}{2}[/itex]

    Also [itex]f(t)= (1+t) \approx f(0) =1[/itex], so that

    \begin{equation}
    I(x)\sim \int^{\delta}_{0} e^{x \frac{\pi}{2}t} dt
    \end{equation}

    after that I tried doing integration by substitution [itex]\tau = x \frac{\pi}{2} t[/itex] but did not get the answer.

    please help me.
     
  2. jcsd
  3. Apr 19, 2014 #2

    Simon Bridge

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    2016 Award

    Laplaces method?
    http://en.wikipedia.org/wiki/Laplace's_method
    ... see "other formulations" and compare with what you did.
    Did you correctly identify the function h(t)?

    Please show your best attempt (working and reasoning) using this method.
    I have a strong feeling about where you went wrong but I don't want to waste my time and yours on a guess.
     
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