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Math Help - Inverse fourier transform using tables

  1. #1
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    Inverse fourier transform using tables

    By using tables find the inverse Fourier transform of \displaystyle e^{-\frac{k^2}{4} + \frac{2}{3} k}.

    The definition of inverse Fourier transform of F(k) in my lecture notes is f(x) = \mathcal{F}^{-1} \{ F(k) \} = \frac{1}{\sqrt{2\pi}} \int^\infty_{-\infty} e^{-ikx} F(k) dk.

    I guess this would involve convolution as \mathcal{F}^{-1} \{ e^{-\frac{k^2}{4}} \} = \sqrt{2} e^{-x^2}. What is \mathcal{F}^{-1} \{ e^{\frac{2}{3} k} \}? How do I get this from the Fourier transforms table?

    The given solution is \sqrt{2} e^{\frac{4}{9}} e^{-i \frac{4}{3} x - x^2}.
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  2. #2
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    Re: Inverse fourier transform using tables

    Quote Originally Posted by math2011 View Post
    By using tables find the inverse Fourier transform of \displaystyle e^{-\frac{k^2}{4} + \frac{2}{3} k}.

    The definition of inverse Fourier transform of F(k) in my lecture notes is f(x) = \mathcal{F}^{-1} \{ F(k) \} = \frac{1}{\sqrt{2\pi}} \int^\infty_{-\infty} e^{-ikx} F(k) dk.

    I guess this would involve convolution as \mathcal{F}^{-1} \{ e^{-\frac{k^2}{4}} \} = \sqrt{2} e^{-x^2}. What is \mathcal{F}^{-1} \{ e^{\frac{2}{3} k} \}? How do I get this from the Fourier transforms table?

    The given solution is \sqrt{2} e^{\frac{4}{9}} e^{-i \frac{4}{3} x - x^2}.
    Complete the square of the exponent then use the appropriate translation theorem and the inverse transform of e^{-w^2}

    CB
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  3. #3
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    Re: Inverse fourier transform using tables

    Thank you. That worked.

    e^{-\frac{k^2}{4} + \frac{2}{3} k} = e^{-(x - \frac{4}{3})^2 + \frac{4}{9}}
    \mathcal{F}^{-1} \{e^{-(x - \frac{4}{3})^2 + \frac{4}{9}} \} = \sqrt{2} e^{\frac{4}{9}} e^{-x^2} e^{-\frac{4}{3} i x}
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