Analytical continuation of moment generating function

Question

Let's say some distribution $F(t)$ has finite moment generating function on an open ball (-R, R).

$M(x) = \sum m_n x^n /n!$

Let's extend $M(x)$ to $M(z)$ on a complex strip $S = \{z| |Re(z)| <R\}$.

I need to prove that $M(z)$ is also analytic on $S$(thus it is an analytic continuation).

What is the easiest (i.e. self-contained) way to do it? I can easily prove that M(z) is bounded, and that $\oint M(z) = 0$ on any path in $S$, but I don't think those are sufficient conditions for $M(z)$ being analytic.

Answer 1

You're just missing one tiny piece of complex analysis to finish your proof, namely Morera's Theorem.