Homework Help: Fourier Transform

1. Feb 20, 2012

fauboca

$g$ is continuous function, $g:[-\pi,\pi]\to\mathbb{R}$

Prove that the Fourier Transform is entire,

$$G(z)=\int_{-\pi}^{\pi}e^{zt}g(t)dt$$

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

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

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

If that is correct, I am also having some trouble at this part as well.

2. Feb 20, 2012

sunjin09

My guess is you can separate the real and imag part of G(z) and show that they satisfy Cauchy-Riemann equations

3. Feb 20, 2012

fauboca

How are the partials done? x and y and neglect t?

4. Feb 20, 2012

sunjin09

sure, t is a dummy variable, you are essentially exchanging differentiation over x,y and integration over t

5. Feb 20, 2012

fauboca

So the C.R. equations are satisfied and that is it then?

6. Feb 27, 2012

fauboca

What can be done to justify slipping differentiation past the integral?

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