${J_{\nu }}^{\text{'}}\left(j_{\nu ,m}\right)\sim {\left(-1\right)}^m\frac{{\left(2/\nu \right)}^{\frac{2}{3}}}{\pi M\left(a_m\right)}{\sum }_{k=0}^{\infty }\frac{{\beta }_k}{{\nu }^{2k/3}}$