10 Bessel Functions Spherical Bessel Functions 10.56 Generating Functions 10.58 Zeros

§10.57 Uniform Asymptotic Expansions for Large Order

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Keywords:: asymptotic approximations for large order, spherical Bessel functions, uniform asymptotic expansions for large order
Notes:: For (10.57.1) use the differentiated form of the first of (10.47.3).
Permalink:: http://dlmf.nist.gov/10.57
See also:: Annotations for Ch.10

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

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, : integer and : complex variable
Referenced by:: §10.57
Permalink:: http://dlmf.nist.gov/10.57.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §10.57 and Ch.10

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

10.56 Generating Functions 10.58 Zeros

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