# §10.57 Uniform Asymptotic Expansions for Large Order

Asymptotic expansions for $\mathsf{j}_{n}\left((n+\tfrac{1}{2})z\right)$, $\mathsf{y}_{n}\left((n+\tfrac{1}{2})z\right)$, ${\mathsf{h}^{(1)}_{n}}\left((n+\tfrac{1}{2})z\right)$, ${\mathsf{h}^{(2)}_{n}}\left((n+\tfrac{1}{2})z\right)$, ${\mathsf{i}^{(1)}_{n}}\left((n+\tfrac{1}{2})z\right)$, and $\mathsf{k}_{n}\left((n+\tfrac{1}{2})z\right)$ 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 ${\mathsf{i}^{(2)}_{n}}\left((n+\tfrac{1}{2})z\right)$ the connection formula (10.47.11) is available.

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

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

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