Suppose $X$ is a random variable and $\psi(t)=E[\exp(itX)]$ is its characteristic function. Let $K(t)$ be the principal value of the logarithm of $\psi(t)$. Suppose further that $E(|X|^{r+2})<\infty$ for some integer $r\geq 0$.
Then, my reading material (the question is self-contained, link included for completeness) claims that the Taylor's Theorm implies $$ K(t)=\sum_{j=0}^{r+2}\kappa_j\frac{(it)^j}{j!}+o(\tau^{r+2}),\quad t\in\mathbb{R}, $$ where $$ \kappa_j=\frac{K^{(j)}(0)}{i^j},\quad j=0,\ldots,r+2, $$ are the cumulants of $X$.
Question: which version of Taylor's Theorem is this and where can I find its rigorous statement (and proof)? (preferred source: textbook)
I only know 2 verions of Taylor's Theorem: one deals with real functions of real variables (univariate or multivariate) and one deals with holomorphic functions. As $K$ may not have all of its derivatives, I don't think the latter applies.