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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(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 $x\rightarrow\infty$.
    => I have tried using the expansion of $I(x)$ in McLaurin series but did not get the answer.
    here,
    \begin{equation}
    h(t)=cos(\frac{\pi(t-1)}{2})
    \end{equation}
    $h(0)= 0$
    $h'(0)= \frac {\pi}{2}$
    Also $f(t)= (1+t) \approx f(0) =1$, 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 $\tau = x \frac{\pi}{2} t$ but did not get the answer.
    please help me.
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  2. #2
    MHF Contributor
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    California
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    Re: Laplace's Method (Integration)

    Quote Originally Posted by grandy View Post
    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 $x\rightarrow\infty$.
    => I have tried using the expansion of $I(x)$ in McLaurin series but did not get the answer.
    here,
    \begin{equation}
    h(t)=cos(\frac{\pi(t-1)}{2})
    \end{equation}
    $h(0)= 0$
    $h'(0)= \frac {\pi}{2}$
    Also $f(t)= (1+t) \approx f(0) =1$, 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 $\tau = x \frac{\pi}{2} t$ but did not get the answer.
    please help me.
    Laplace's Method says

    $\Large I=\displaystyle{\int_a^b}e^{-x g(t)}h(t) ~dt \approx e^{-x g(\hat{t})}\sqrt{\dfrac{2 \pi}{x g''(\hat{t})}}$

    here

    $g(t)=\cos\left(\dfrac{\pi (t-1)}{2}\right)$

    $h(t)=1+t$

    $\hat{t}=1$

    see if you can finish from here.
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