Notation used is taken from Gallager's text on stochastic processes.
For a random variable $X$, let $g_X(r)=\mathbb{E}[\exp(rX)]$ be its moment generating function where $r$ is a real number. Similarly, let $g_X(i\theta) = \mathbb{E}[\exp(i\theta X)]$ where $\theta$ is a real number and $i=\sqrt{-1}$.
In a footnote, the author warned that the notation is slightly dangerous because $g_X(i\theta)$ cannot be simply obtained from the MGF with $r=i\theta$.
So far, I have only used MGFs and characteristic functions for (jointly) Gaussian random variables, and for that class of rvs, the "substitution" seemed to be work, e.g. for $X\sim \mathcal{N}(0,\sigma^2)$, the MGF is $g_X(r)=\exp(r^2 \sigma^2 /2)$ and the characteristic function is $g_X(i\theta)=\exp(-\theta^2 \sigma^2 /2)$.
My question is: for which random variables will using $r=i\theta$ on MGFs correctly give the characteristic function?
Best Answer
Setting $r=i\theta$ has the effect of attempting an analytic continuation of the MGF from real arguments to imaginary ones. Similarly, setting $\theta=-ir$ tries to analytically continue the characteristic function from real arguments to imaginary ones. So it all works iff the characteristic function is analytic on $\Bbb C$. For example, it works for Poisson ($\Bbb E\exp i\theta X=e^{\lambda(\exp i\theta-1)}$) but not standard Cauchy ($\Bbb E\exp i\theta X=e^{-|\theta|}$).