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$$

G(z)=\int_{-\pi}^{\pi}e^{zt}g(t)dt, \quad \text{for all} \ z\in\mathbb{C}.

$$

Prove that $G(z)$ is an entire function.

That means $G$ has to have no singularities, but other than that I am lost. We have to continuous functions multiplied together but then what?