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Math Help - Laplace's Method Integration

  1. #1
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    Laplace's Method Integration

    Consider the integral
    \begin{equation}
    I_n(x)=\int^{2}_{1} (log_{e}t) e^{-x(t-1)^{n}}dt
    \end{equation}
    Use Laplace's Method to show that
    \begin{equation}
    I_n(x) \sim \frac{1}{nx^\frac{2}{n}} \int^{\infty}_{0} \tau^{\frac{2-n}{n}} e^{-\tau} d\tau \end{equation}
    as $x\rightarrow\infty$.
    where $0<n\leq2$. Hence find the leading order behaviour of $I_{1}(x)$. and $I_{2}(x)$ as $x\rightarrow \infty$.
    =>
    Its really difficult question for me.
    Here,
    $g(t) = -(t-1)^{n}$ has the maximum at $t=0$
    but $h(t)= log_{e}t$ at $t=0$
    $h(0)=0$.
    so I can not go any further. PLEASE HELP ME.
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  2. #2
    MHF Contributor

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    Re: Laplace's Method Integration

    First can you give the precise statement of "Lapace's method"?
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