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Fourier Transform

  1. Feb 20, 2012 #1
    [itex]g[/itex] is continuous function, [itex]g:[-\pi,\pi]\to\mathbb{R}[/itex]

    Prove that the Fourier Transform is entire,

    [tex]
    G(z)=\int_{-\pi}^{\pi}e^{zt}g(t)dt
    [/tex]

    So,
    [tex]G'(z) = \int_{-\pi}^{\pi}te^{zt}g(t)dt=H(z)[/tex].

    Then I need to show that [itex]G(z)[/itex] differentiable for each [itex]z_0\in\mathbb{C}[/itex].

    I need to show [itex]\left|\frac{G(z)-G(z_0)}{z-z_0}-H(z_0)\right| < \epsilon[/itex] whenever [itex]0<|z-z_0|<\delta[/itex], correct?

    If that is correct, I am also having some trouble at this part as well.
     
  2. jcsd
  3. Feb 20, 2012 #2
    My guess is you can separate the real and imag part of G(z) and show that they satisfy Cauchy-Riemann equations
     
  4. Feb 20, 2012 #3
    How are the partials done? x and y and neglect t?
     
  5. Feb 20, 2012 #4
    sure, t is a dummy variable, you are essentially exchanging differentiation over x,y and integration over t
     
  6. Feb 20, 2012 #5
    So the C.R. equations are satisfied and that is it then?
     
  7. Feb 27, 2012 #6
    What can be done to justify slipping differentiation past the integral?

    How can I show the partials are continuous at this point?
     
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