\[J_{\nu}\left(\nu z\right)=\left(\frac{4\zeta}{1-z^{2}}\right)^{\frac{1}{4}}%
\left(\frac{\operatorname{Ai}\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{1%
}{3}}}\left(\sum_{k=0}^{\ell}\frac{A_{k}(\zeta)}{\nu^{2k}}+O\left(\frac{1}{%
\zeta^{3\ell+3}}\right)\right)+\frac{\operatorname{Ai}'\left(\nu^{\frac{2}{3}}%
\zeta\right)}{\nu^{\frac{5}{3}}}\left(\sum_{k=0}^{\ell-1}\frac{B_{k}(\zeta)}{%
\nu^{2k}}+O\left(\frac{1}{\zeta^{3\ell+1}}\right)\right)\right),\]