I just calculated the characteristic function of a normally distributed variable. However, what's weird is that I used the assumption that a noormally distributed variable can have a complex mean, even if I'm just integrating over the real line.
Perhaps I only accidentally got the right answer?
I got to this point:
$$f(L) =e^{iL\mu-0.5 L^2\sigma^2} \int \frac {1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-(\mu+iL\sigma^2))^2}{2\sigma^2}} \, dx = e^{iL\mu-0.5 L^2\sigma^2}$$
Does this make sense? if so, why would it, given that we are only integrating over the real line? Yet the answer is correct.