This feels a bit silly, but I can't seem to find a nice reference. I have a weakly converging sequence of random variables $X_n \Rightarrow X$. The characteristic functions $\phi_n(s) := \mathbb{E}[e^{isX_n}]$ are analytic in a fixed complex neighborhood $N$ of $0$ and converge uniformly on compact subsets of $N$ to an analytic function $\psi(s)$ on $N$.
By Levy continuity, $\phi(s) := \mathbb{E}[e^{isX}] = \psi(s)$ for $\mathbb{R} \cap N$. I want to conclude that $$\psi(s) = \mathbb{E}[e^{isX}] \qquad\text{for all }s \in N.$$ Can I do so? Moreover, is it written down somewhere? It just seems like such a natural thing to want, but none of the sources I've looked at (e.g. Billingsley) talk about it, so I'm probably looking in the wrong places.
This follows from Lukacs, Characteristic Functions, 2nd Edition (1970), Thm. 7.1.1 and the discussion preceding it (p. 191-193). As you would expect, Lukacs shows that if a characteristic function agrees with an analytic function in a neighborhood of the origin, then $\mathbb{E}[e^{izX}]$ agrees with that analytic function in a strip $-\alpha < \mathrm{Im}(z) < \beta$, which is exactly my desired conclusion.