# Characteristic funtion of RV

so a charateristic function of a RV is complex valued funtion. from my lecture, the distribution funtion of a Random variable is not always "well behaved", may not have a density etc. A charateristic function on the other had is "well behave".
What i dont understand is, is that the only reason we use it ?
how is it actually derived, why does it have to be complex valued .
this is the definition i'm given $$\phi(t) = \mathbb{E}(e^{itX})$$
how is this actually derived, is somewhere where i can find the proof ?

mathman
You need to clarify your question. Definitions aren't derived.

As far as usage, the simplest example is deriving the distribution function a sum of independent random variables. The characteristic function of the sum is the product of the characteristic functions of the individual variables.