The following asymptotic expression for the Bessel function holds according to Abramowitz and Stegun:
$$ J_\nu(z) = \sqrt{\frac{2}{\pi z}} \left(\cos\left(z - \frac{\nu\pi}{2} - \frac{\pi}{4}\right) + \mathrm{e}^{|\operatorname{Im}(z)|} O(|z|^{-1})\right)$$
Does equality just straight up hold here, or are there some size conditions I need to worry about?