On the first page of this document the expansion
\begin{align} e^{ix} = (1+ix)e^{-\frac{x^2}{2} + r(x)} \end{align}
is given,where $|r(x)| \leq |x^3|$ for all real $x$.
In the original proof by Mcleish the statement above is valid only for $|x| < 1$ and is found by taking the log of both sides of the equation and taking a second order expansion of $\ln(1+ix)$. As far as I understand this cannot be used to demonstrate for all real $x$ as $\ln(x)$ is not complex analytic everywhere that is necessary. How then can this proved for all real $x$?