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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.