Digital Library of Mathematical Functions
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§10.57 Uniform Asymptotic Expansions for Large Order

Asymptotic expansions for jn((n+12)z), yn((n+12)z), hn(1)((n+12)z), hn(2)((n+12)z), in(1)((n+12)z), and kn((n+12)z) as n 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 in(2)((n+12)z) the connection formula (10.47.11) is available.

For the corresponding expansion for jn((n+12)z) use

10.57.1 jn((n+12)z)=π12((2n+1)z)12Jn+12((n+12)z)-π12((2n+1)z)32Jn+12((n+12)z).

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