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Math Help - Trouble with the integration part of an inverse fourier transform

  1. #1
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    Trouble with the integration part of an inverse fourier transform

    I'm looking for some help on the following inverse fourier transform if possible as I don't see how to proceed to prove the required answer. I also am not sure how to make it look like proper maths on this web page so please bear with me I'll try and make it look as understandable as possible:

    F^-1 [2pi*delta(w-(w0))] = e^i(w0)t

    where w0 is a constant and by delta I mean 'change in'. I think the rest is ok.

    The only bit I've managed to do so far is to cancel the 2pi in the function with the 1/2pi constant attached to the inverse transform equation... I'm not sure how to integrate the rest.

    Any help would be much appreciated
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  2. #2
    Super Member girdav's Avatar
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    Re: Trouble with the integration part of an inverse fourier transform

    If \varphi\in S(\mathbb R), where S(\mathbb R) denotes the Schwartz space, since \delta_{\omega_0} has a compact support, \delta_{\omega_0}\in S'(\mathbb R) and we can write \langle F^{-1}(2\pi\delta_{\omega_0}),\varphi\rangle_{S' (\mathbb{R}),S(\mathbb{R})} =\langle 2\pi\delta_{\omega_0},F^{-1}(\varphi)\rangle_{S'(\mathbb{R}),S(\mathbb{R})}.
    Now, you have to notice that 2\pi F^{-1}(\varphi) (x) = \int_{\mathbb R}e^{ixt}\varphi(t)dt, and applying \delta_{\omga_0} to this function we find \langle F^{-1}(2\pi\delta_{\omega_0}),\varphi\rangle_{S' (\mathbb{R}),S(\mathbb{R})}  = \int_{\mathbb R}e^{i\omega_0 t}\varphi(t)dt, and we recognize the distribution which is associated to the function t\mapsto e^{i\omega_0 t}.
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