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Sin/cos integrals multiplying results (fourier transform).

  1. Feb 28, 2014 #1
    Okay, I am trying to determine the fourier transform of cos (2[itex]\pi[/itex]x)=f(x)

    Where F(k)=[itex]^{\infty}_{\infty}[/itex][itex]\int[/itex]f(x)exp[itex]^{-ikx}[/itex] dx,

    So I use eulers relation to express the exponential term in terms of cos and sin, and then I want to use sin/cos multiplication integral results, such as:

    [itex]^{\infty}_{-\infty}[/itex][itex]\int[/itex] cos(nx)cos(mx) dx =[itex]\pi[/itex] if m=n≠0
    = 2[itex]\pi[/itex] if m=n=0
    =0 if m≠n


    - But these are only defined for limits [itex]\pm[/itex][itex]\pi[/itex].
    So my question is , what are these results for [itex]\pm[/itex][itex]\infty[/itex].

    Is there a obvious natural extension?

    Many thanks for any assistance !
     
    Last edited: Feb 28, 2014
  2. jcsd
  3. Feb 28, 2014 #2

    LCKurtz

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    One of the requirements for a Fourier transform of a function ##f## to exist is$$
    \int_{-\infty}^\infty |f(x)|~dx$$converge. Sines and cosines don't satisfy that. Apparently there is some sense in which it can be expressed with delta functions.
     
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