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10 Bessel FunctionsSpherical Bessel Functions

§10.57 Uniform Asymptotic Expansions for Large Order

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Notes:
For (10.57.1) use the differentiated form of the first of (10.47.3).
Keywords:
spherical Bessel functions
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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^{{\prime}}}\!\left((n+\tfrac{1}{2})z\right) use

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

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