# §10.57 Uniform Asymptotic Expansions for Large Order

Asymptotic expansions for $\mathop{\mathsf{j}_{n}\/}\nolimits\!\left((n+\tfrac{1}{2})z\right)$, $\mathop{\mathsf{y}_{n}\/}\nolimits\!\left((n+\tfrac{1}{2})z\right)$, $\mathop{{\mathsf{h}^{(1)}_{n}}\/}\nolimits\!\left((n+\tfrac{1}{2})z\right)$, $\mathop{{\mathsf{h}^{(2)}_{n}}\/}\nolimits\!\left((n+\tfrac{1}{2})z\right)$, $\mathop{{\mathsf{i}^{(1)}_{n}}\/}\nolimits\!\left((n+\tfrac{1}{2})z\right)$, and $\mathop{\mathsf{k}_{n}\/}\nolimits\!\left((n+\tfrac{1}{2})z\right)$ as $n\to\infty$ that are uniform with respect to $z$ can be obtained from the results given in §§10.20 and 10.41 by use of the definitions (10.47.3)–(10.47.7) and (10.47.9). Subsequently, for $\mathop{{\mathsf{i}^{(2)}_{n}}\/}\nolimits\!\left((n+\tfrac{1}{2})z\right)$ the connection formula (10.47.11) is available.

For the corresponding expansion for $\mathop{\mathsf{j}_{n}\/}\nolimits'\!\left((n+\tfrac{1}{2})z\right)$ use

 10.57.1 $\mathop{\mathsf{j}_{n}\/}\nolimits'\!\left((n+\tfrac{1}{2})z\right)=\frac{\pi^% {\frac{1}{2}}}{((2n+1)z)^{\frac{1}{2}}}\mathop{J_{n+\frac{1}{2}}\/}\nolimits'% \!\left((n+\tfrac{1}{2})z\right)-\frac{\pi^{\frac{1}{2}}}{((2n+1)z)^{\frac{3}{% 2}}}\mathop{J_{n+\frac{1}{2}}\/}\nolimits\!\left((n+\tfrac{1}{2})z\right).$

Similarly for the expansions of the derivatives of the other six functions.