Fourier Transform

  • Thread starter fauboca
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  • #1
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[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.
 

Answers and Replies

  • #2
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My guess is you can separate the real and imag part of G(z) and show that they satisfy Cauchy-Riemann equations
 
  • #3
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My guess is you can separate the real and imag part of G(z) and show that they satisfy Cauchy-Riemann equations
How are the partials done? x and y and neglect t?
 
  • #4
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How are the partials done? x and y and neglect t?
sure, t is a dummy variable, you are essentially exchanging differentiation over x,y and integration over t
 
  • #5
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sure, t is a dummy variable, you are essentially exchanging differentiation over x,y and integration over t
So the C.R. equations are satisfied and that is it then?
 
  • #6
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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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