Proving Fourier Transform is Entire

[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.

 

[sunjin09]

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

 

[fauboca]

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?

 

[sunjin09]

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

 

[fauboca]

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?

 

[fauboca]

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

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