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Fourier transform integration using well-known result watch

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    F denotes a forward fourier transform, the variables I'm transforming between are x and k

    So first of all I note I am given a result for a forward fourier transform and need to use it for the inverse one.

    The result I am given to use, written out is :

     \int e^{-ikx} e^{-\alpha x^{2}} dx = \frac{1}{2\alpha} e^{-k^{2}/4\alpha}

    I note that  F(k) C(t) gives me a function of k, so I apply  F^{-1} on this I get a function of x.

    My thoughts are I'm looking to change signs in my exponential terms so that it is , effectively a forward transform and then use the result, so just thinking of it as a integration result rather than a particular fourier transform.

    However if I do this my exponential terms are:

     e^{-(-ikx+\alpha k^{2})}
    , I don't know how I can then apply the result, completing the square looks like the only candidate to me , but this seems like it will be too scrappy, in particular with 'i' terms, how do I deal with this  e^{ikx} term, if I'm on the right lines, is completion of the square necessary or is there some other approach to being able to use this result ?

    Many thanks in advance.
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    (Original post by TeeEm)
    I think there similar if not identical questions here
    http://www.madasmaths.com/archive/ma..._transform.pdf
    q20?

    doesn't actually answer my question above though, just shows me how to integrate.

    What I've been asked to do should probably be simpler, but I'm more interested in the logic I'm missing relevant to the OP.

    edit: apologies 24 is the relevant one I think, still it does cos and sin intergration rather than using the result of a forward gaussian transform, i.e. rather than using q21
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    (Original post by xfootiecrazeesarax)
    q20?

    doesn't actually answer my question above though, just shows me how to integrate.

    What I've been asked to do should probably be simpler, but I'm more interested in the logic I'm missing relevant to the OP.

    edit: apologies 24 is the relevant one I think, still it does cos and sin intergration rather than using the result of a forward gaussian transform, i.e. rather than using q21
    I'm reasonably confident that the result you've been given is off by a factor of \sqrt{2\alpha}. Also the factor of \sqrt{2\pi} in the final result is dependent on how you define your Fourier transform, and the inverse.
    Here are a couple of bits of evidence to back me up:
    Wolfram.
    Page 3 of this.
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    (Original post by joostan)
    I'm reasonably confident that the result you've been given is off by a factor of \sqrt{2\alpha}. Also the factor of \sqrt{2\pi} in the final result is dependent on how you define your Fourier transform, and the inverse.
    Here are a couple of bits of evidence to back me up:
    Wolfram.
    Page 3 of this.
    okay cheers, easy parts to pick up after, my main concern is the logic, any tips on this?
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    (Original post by xfootiecrazeesarax)
    okay cheers, easy parts to pick up after, my main concern is the logic, any tips on this?
    Well I'd just choose a value for \alpha that puts the integral into the desired form.
    Then use an inverse fourier transform, using the given result to obtain the desired answer.
    There's not much in the way of logic to it . . .
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    (Original post by joostan)
    Well I'd just choose a value for \alpha that puts the integral into the desired form.
    Then use an inverse fourier transform, using the given result to obtain the desired answer.
    There's not much in the way of logic to it . . .
    okay I think I'm being stupid,

    but then alpha= Dt, but I have the exponent power as -(-ikx -alpha k^{2}) rather than (-ikx-alpha k^{2}) so I have no idea how can apply the result given...?

    ta
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    (Original post by xfootiecrazeesarax)
    okay I think I'm being stupid,

    but then alpha= Dt, but I have the exponent power as -(-ikx -alpha k^{2}) rather than (-ikx-alpha k^{2}) so I have no idea how can apply the result given...?

    ta
    It would help if at some point you could write down exactly what integral you are trying to evaluate - the question is cut off, so I'm not 100% certain you're trying to find \displaystyle \frac{Q}{2\pi}\int_{-\infty}^\infty e^{ikx} e^{-Dk^2t} \,dk or something else.

    I'm not seeing how you can possibly have the exponent being -(-ikx -\alpha k^2) (which equals  ikx + \alpha k^2 and therefore has a positive k^2 term and is divergent whene you integrate w.r.t. k)

    But from other posts you seem to be bothered by the fact that you have e^{ikx} instead of e^{-ikx} and I suspect you're tying yourself in knots trying to resolve this.

    It actually doesn't matter about the sign of this term - this doesn't really even require calculation: you know what F(k) is, so putting -k into the same formula you can see F(-k) is the same thing.

    [If you want to see "why" this holds, you can see this either by doing a substittuion (of form \lambda = -k) or by observing that e^ikx e^(-ax^2)= cos kx e^(-ax^2) + i sin kx e^(-ax^2) The cos term is even (so is the same if you change the sign of k), and the sin term is odd, so vanishes when you integrate over a symmetric region (and so is the same if you change the sign of k. But you don't actually need to do this - it's implicit in the formula you're given].
 
 
 
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