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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.