§10.57 Uniform Asymptotic Expansions for Large Order

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

For the corresponding expansion for ${\mathsf{j}}_n^{\prime }\left((n+\frac{1}{2})z\right)$ use

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