$\begin{array}{rcl}J{\text{'}}_v(x)& \text{=}& N_v\cos {\varphi }_v\\ Y{\text{'}}_v(x)& \text{=}& N_v\sin {\varphi }_v\\ N_v^2& \approx & \frac{2}{\pi x}\left\{1-\frac{1}{2}\frac{\mu -3}{{(\mathrm{2x})}^2}-\frac{1}{8}\frac{(\mu -1)(\mu -45)}{{(\mathrm{2x})}^4}\right\}\\ {\varphi }_v& \approx & x-(\frac{1}{2}v-\frac{1}{4})\pi +\frac{\mu +3}{2(\mathrm{4x})}+\frac{{\mu }^2+46\mu -63}{6{(\mathrm{4x})}^3}+\frac{{\mu }^3+185{\mu }^2-2053\mu +1899}{5{(\mathrm{4x})}^5}\\ \mu & =& {\mathrm{4v}}^2\end{array}$