this is a general question. Is there a general way to expand the Bessel Function $J_1(z)$ when $z\in \mathbb{C}$ and when z is large? Or in other words, what is the asymptotic expansion of $J_1(z)$? I think it is sufficient to just use $$ J_1(z)\sim \frac{1}{\sqrt{2\pi z}} e^{-i(z-\pi/4)},\quad 0<\arg z<\pi. $$ Is this correct? The general power series expansion is given by $$ J_1(z)=\frac{z}{2}\sum_{k\geq 0} \frac{(-1)^k}{k!\Gamma(k+2)} \left(\frac{z}{2}\right)^{2k}. $$ For $z\to 0$, it is sufficient to keep the lower order terms, thus we can write something like $$ J_1(z)\approx \frac{z}{2}-\frac{z^3}{16}. $$
Thanks, see here for more details of what I just wrote: https://en.wikipedia.org/wiki/Bessel_function