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Math Help - problem on Fourier transforms

  1. #1
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    problem on Fourier transforms

    I need some help with this also:

    Prove that :

    FourierTransform[e^(-ax^2)]=(sqrt[Pi/a]) e^(-ω^2/4a)
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  2. #2
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    Quote Originally Posted by jmytil View Post
    I need some help with this also:

    Prove that :

    FourierTransform[e^(-ax^2)]=(sqrt[Pi/a]) e^(-ω^2/4a)
    Put

    <br />
f(x)=e^{-ax^2}<br />

    Differentiate:

    <br />
\frac{df}{dx}=-2 a x f(x)<br />

    Now take FTs of both sides and apply the derivative rules to the forward
    and backwards transform:

    <br />
i \omega F(\omega) = -2 a i\frac{dF}{d\omega}<br />

    so we have F(\omega) satisfies the ODE:

    <br />
\omega F(\omega) = -2 a\frac{dF}{d\omega}<br />

    and all one now has to do is show that the suggested form for F(\omega) satisfies
    this equation (or you could just solve the equation - it's of variables separable type).

    (there is at least one other way of doing it, but that requires Cauchy's integral theorem from complex analysis)

    RonL
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