Are there general conditions under which the moment generating function $M(t)$, $t\in\mathbb{R}$, and the characteristic function $\phi(t)$, $t\in\mathbb{R}$ can be extended to a complex argument. Please provide a citation.
I can find the following results on Wikipedia:
1. $\phi(t) =M(it)$, $t\in\mathbb{R}$. But under what conditions is this true? Is it true for any mgf? Is there a citation?
2. $\phi(-it) =M(t)$, $t\in\mathbb{R}$ for any mgf. But is there a general condition for the existence of a mgf?
Let $f$ be the probality distribution of a real random variable $X$. $$\phi(t)=\Bbb{E}[e^{iXt}] = \int_{-\infty}^\infty e^{ixt} f(x)dx, t\in \Bbb{R}$$ is bounded, continuous and if $\Bbb{E}[X^{2m}]$ exists then $\phi$ is $C^{2m}(\Bbb{R})$ and $$\phi^{(n)}(0) = i^n \Bbb{E}[X^{n}], n \le 2m$$
If also $\Bbb{E}[e^{a|X|}]$ exists for some $a > 0$ then $$\phi(z)= \int_{-\infty}^\infty e^{ixz} f(x)dx,z \in \Bbb{C},|\Im(z) | < a$$ converges and is analytic and $$\phi(z) = \sum_{m=0}^\infty \frac{1}{m!}z^m i^m\Bbb{E}[X^m]$$
The intermediate but more complicated case is when $\phi(t)$ is analytic near $t=0$ but $ \int_{-\infty}^\infty e^{ax} f(x)dx$ doesn't converge.