I was calculating the characteristic function of a random variable, and I got stuck here at this point
$$\int_{a}^{b} \exp\left(it\log(\alpha)\right)\exp\left(-\frac{1}{2}\left(\frac{\alpha-\mu}{\sigma}\right)^2\right)\mathrm{d}\alpha$$
I really don't know how to proceed from this point, and I would apreciate it if you guys could give me some hints.
Note that: $$\exp(it\log\alpha)=\alpha^{it}$$
You can then perform a change of variables:$$y=\frac{1}{2}\left(\frac{\alpha-\mu}{\sigma}\right)^2$$
To get an integral of the form: $$\frac{1}{2}\int\left(\sqrt{2y\sigma^2}+\mu\right)^{it-1}e^{-y}dy$$ When $\mu$ is zero, the solution is given directly by a difference between Incomplete Gamma Function. When $\mu\neq0$, we might need some more massaging.