If we have a Bessel function of the first kind and the $m$-th order as $J_{m}(x+i\epsilon x)$, where $m$ is integer, $x, \epsilon$ are real and $\epsilon$ is a small parameter ($0<\epsilon\ll 1$), can we expand this in Taylor series and write
$$ J_{m}(x+i\epsilon x) \approx J_{m}(x)+i\epsilon x J^{'}_{m}(x)-\frac{(\epsilon x)^{2}}{2}J^{''}_{m}(x)+\cdots \ \ \ ?$$
I just want to double check if this is a correctly written Taylor expansion, since the coefficients here contain $x$?