Domain of characteristic function

Question

Hey if $X$ is random variable, then what is the domain of characteristic function of $X$? In many books there is written that the domain is $\mathbb{R}$, but sometimes authors takes value of characteristic function at point $-i$ (which is complex number) when they caculate $\mathbb{E}(e^X)$. So what is the domain and if complex, what are the differences between these two domains?

Answer 1

Continuing my comment, the "approach" you mentioned in your post is invalid for some very common characteristic function.

It is well known that the c.f. of the standard exponential distribution $f(x) = e^{-x}, x > 0$ is

\begin{align*} \phi(t) = \frac{1}{1 - it}, t \in \mathbb{R}. \end{align*}

When you literally plugging $t = -i$, the denominator will be $0$, hence inconclusive. In fact, the expectation of $e^X$ is not a finite number for this case.