Characteristic function to pdf

Can someone give me an example of solving an integral of an inverse characteristic function? For example, let's say we have the characteristic function of a gaussian. How would I recover the probability distribution from that? I can't seem to solve the integral.

Ultimately, I'm trying to use a moment generating function to recover some probability distribution from that. I want to create an arbitrary series of moments (subject to some constraints) and then get a pdf from that. I just can't remember enough complex maths to solve the integrals! Thanks.

(I'm cross-posting the question to the Advanced Topics forum...)