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Thread: help for asymptotic expansion

  1. #1
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    help for asymptotic expansion

    Hi im new to this forum and was wondering if someone could help me with the following problem


    use a suitable series expansion to find an asymptotic expansion as x approaches infinity for

    the integral from 0 to infinity of [ exp(-x(t^2))sin(t) ]

    i tried expanding the sine part and integrating term by term. i got stuck doing integrating term by term. i dont know how to do integral im not sure if i should use a different expansion?


    thanks in advance
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by jules View Post
    Hi im new to this forum and was wondering if someone could help me with the following problem


    use a suitable series expansion to find an asymptotic expansion as x approaches infinity for

    the integral from 0 to infinity of [ exp(-x(t^2))sin(t) ]

    i tried expanding the sine part and integrating term by term. i got stuck doing integrating term by term. i dont know how to do integral im not sure if i should use a different expansion?


    thanks in advance
    We want an expansion of

    $\displaystyle
    I(x) = \int_0^{\infty} \exp(-x t^2)\sin(t) dt
    $

    for large positive $\displaystyle x$.

    First put $\displaystyle u=\sqrt{x}t$, then the integral becomes:

    $\displaystyle
    I(x) = \frac{1}{\sqrt{x}}\int_0^{\infty} \exp(- u^2)\sin(u/\sqrt{x}) du
    $

    Now as $\displaystyle x$ is large positive and $\displaystyle \exp(-u^2)$ decays rapidly for large $\displaystyle u$ we can expand the $\displaystyle \sin(u/\sqrt{x})$ to get:

    $\displaystyle
    I(x) = \frac{1}{\sqrt{x}}\int_0^{\infty} \exp(- u^2)\left[ \sum_0^{\infty}\frac{(u/\sqrt{x})^{2n+1}(-1)^{n}}{(2n+1)!} \right] du
    $

    Hopefully you can take it from here.

    RonL
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  3. #3
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    Hi
    Thanks that helped. Though im still not clear on how to do that integral??
    Thanks
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by jules View Post
    Hi
    Thanks that helped. Though im still not clear on how to do that integral??
    Thanks
    Standard integral:

    $\displaystyle
    \int_0^{\infty} \exp(-u^2) ~u^k du = \frac{1}{2} \Gamma\left(\frac{2k+1}{2}\right)
    $

    In the case of the power you have the $\displaystyle \Gamma$ will simplify to a factorial.

    RonL
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