Suppose you have a continuous function:
$$\phi:[0,1]\rightarrow \mathbb{C}$$
define the complex function:
$$f(z)=\int_0^1\phi(t)e^{itz}dt$$
prove that it is entire and calculate it's Taylor expansion centered at $z=0$. Honestly I don't know where to start, I think I have to apply the theorem of holomorphy of a parametric integral but I don't understand how.
Also, how can I apply those results to the sequence of functions:
$$f_n(z)=\int_0^n \sqrt{t}e^{-tz}dt$$
It's obviously entire as the derivative is $$ \int_0^1 ite^{izt} \phi(t) \, dt$$ which exists (is convergent) for all $z$ as the integrand is bounded. The Taylor series is obtained by differentiating under the integral sign: the $n$th derivative is $$ \int_0^1 (it)^n \phi(t)\, dt.$$