My self-study in measure and probability theory as finally brought me to the subject of characteristic functions, and I have not handled these in the past with any rigor at all, so all of this is somewhat daunting.
Problem: To find explicitly the characteristic function for each of these situations:
$X$~Poisson$(\lambda)$
$Y$~Gamma$(\alpha,\beta)$
$Z$~$U_1+U_2+...+U_n$ where each $U_n$ is ~Uniform(-1,1) (additionally, can this example be used in Levy's inversion formula if $n\ge 2$ to have a completely real-valued integral?
Work/attempts at solution
For $X$:
$\phi_X(t)=E[e^{itX}]=\sum[e^{it}]^xe^{-\lambda}\frac1{x!}\lambda^x=e^{-\lambda}\sum_{x=0}^{\infty}\frac{(\lambda e^{it})^x}{x!}=e^{-\lambda}e^{\lambda e^{it}}=e^{\lambda e^{it-1}}$
I'm not completely sure on this, but it seems reasonable.
I am pretty stuck on the gamma distribution and am largely clueless on the sum of uniform RVs part.